Calculus 2 Fundamentals - Definite Integrals (Integration) | Tanmay Varshney | Skillshare

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Calculus 2 Fundamentals - Definite Integrals (Integration)

teacher avatar Tanmay Varshney, Software Developer, Tech Educator

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Lessons in This Class

12 Lessons (1h 41m)
    • 1. Introduction to Definite Integrals

    • 2. The Concept

    • 3. Riemann Sum

    • 4. Fundamental Theorem of Calculus

    • 5. Fundamental Theorem of Calculus - Examples

    • 6. The Substitution Rule

    • 7. The Substitution Rule - Example

    • 8. Properties of Definite Integrals - Part 1

    • 9. Properties of Definite Integrals - Part 2

    • 10. Properties of Definite Integrals - Examples

    • 11. Miscellaneous Examples - 1

    • 12. Miscellaneous Examples - 2

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About This Class

About this course:

This course focuses on all the fundamental concepts of Integral calculus - Definite Integrals.

You will crush the basics of The Definite Integrals in this series.

After completing this course, you will be thoroughly prepared to learn the applications of integration like finding the area under the curve, calculating arc length, the surface area of complex functions, and many more.


What is Integral Calculus?

Integral Calculus or integration is all about joining smaller pieces to find the total. 

It is actually the opposite of Differential Calculus.


Practical Applications: 

Integral Calculus has numerous applications in the fields of:

  1. Physics: to calculate the center of mass, center of gravity, moment of inertia, etc

  2. Chemistry: to determine the rate of a chemical reaction

  3. Medical Sciences

  4. Modelling Electronic Circuits 

  5. Predicting Stock Market prices and many more

Hence, it is everywhere.

Course content:

This course has been designed for you to cover all the topics of the  Definite integral in an exhaustive manner.

This course is categorized into:

  1. Video lectures covering the core concepts of definite integral calculus and varied examples.

  2. Miscellaneous Examples to crush the basics even further.

  3. Practice Worksheets to let you crush the learned concepts even further.

Meet Your Teacher

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Tanmay Varshney

Software Developer, Tech Educator


I am a Senior Software Engineer with vast experience of working in top tech giant companies.
I have more than 6 years of industry and teaching experience in domains like:

1. Designing scalable architecture for complex and distributed systems.

2. Developing components in a system across the full stack.

3. Solving complex data structures and algorithms related problems.

These are the major skills needed to be a good software developer who can excel in any tech company easily. I am really passionate about sharing my knowledge and expertise with you.

Thus, I am on board to create awesome technical courses on Skillshare based on my expertise which can be understood in the simplest manner.

Come, join me in this learning adventure!! I will... See full profile

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1. Introduction to Definite Integrals: in this section, you will learn about the definite in eagle. First of all, it was start with the explanation for the concept off definite in Eagle. Then we will discuss how definite in Eagle can be computed as the limit off. Three months. Um, with England the most important tour, um, the fundamental totem off helpless and see it in action. After this, we will see how the substitution rule behaves for different in eagles. You will then be introduced to the common properties off definite in people's. After covering the about topics will practice a variety off challenging examples in the miscellaneous example section. At the end, it will revise all the learned and separates off definite and eagle in the revision at a glance section. So let's Junin to master this section. 2. The Concept: but this video we will start with a new section that is definite integral. A basically definitely legal means indefinite in giggle with limits. Okay, Now suppose we have a continuous function ffx defined on an interval a Toby. So, you know, we were in the previous section for indefinite, integral, even know not taking into account any specific values off X. But from here, let's say we have a country dysfunction from the final, an interval from A to B so we can define the different in giggle as the sign area off the VI gin mounted. But the draft off ffx the x axis and the vertical lines intersecting the X axis at points A and B, that means we have we're putting limits on the integrity and I signed area We mean the area with respect to X Texas. Okay, So the definite in the middle is the sign area off the region bounded by the graph off ffx the X axis and the points a andi So the area bone between all this This is the notion behind the definite integral. So if the bound area comes below the X axis definitely Needle is considered as negative. So the bone video is below X axis. The definite integral is considered as negative and other ways. If the bone media comes above the X axis, then the definite integral is considered as positive. Right now, Formally, a definite integral is defined as in giggle from two points. Let's say it, Toby and fix gigs. So this part is the same Onley. These two limits A and B are being added in the different integral where a is called the lower limit. Oh, off the integral on B is called the upper limit off the integral right. So these values A and B are put at the lure and the upper and respectively off the integral sign. I'd also this ffx This should be integral now. What do we mean by integral Indictable means that the function F X must be continuous over the interval from A to B. So this means the function should be continuous from a Toby, including the points A and B. So this is a close interval from A to B late. Now there are two ways to evaluate definitely eagle off a function and then in trouble. A Toby, it can be evaluated as limit offer some. And the second is if it has an entered a weirdo, pull it and giggle Capital left in this in double it Toby, then its value is the difference between the values off this capital. If that is the degenerative at the points me. And so that is F b minus f a. So these are the two methods to calculate the definite integral, and we will be considering both of the methods in the jail in the upcoming videos. Especially, we will be sticking to the second matter mostly in our course. So this was it for this video stadium for the upcoming videos. 3. Riemann Sum: In this particular video, you will see how they well with definite integral as the limit off. Three months. Um, basically definite. And eagles represent the area under the terrible for function, and Riemann sums help us approximate surge areas. So to clarify how this actually works, let's consider on Dennis function F X, defining the clothes and double A to B and assuming that affects, why is it going to FX is always greater than zero? The falling rough depicts ffx so another in Diggle off FX is the area off the region bound by this girl's bicycle toe. FX with the XX is from points a B so that this area is represented by the region a B C D. In this guff. So the entire region lying between a Toby is under three months. Um, we divided and do an equal seven devils given by ex not to excellent X one to x two and at at the art point, we say x r minus one to ex are on Finally from ex N minus one to exit. So these are an equal $7 mitt out off all the away. $7 are off of it. let's say h right So the bit off its amendable is etch and there and equal seven billets an equal seven of us off that at each like so this is our consideration now, since each is off with EJ and the starting point that is X not is equal to a so it's not is equal to a that means x one. The next point of consideration will be at a weight off its track That will be a plus. Etch So XTO will be similarly a pless to which and so on xn minus one. This will be a plus and minus one edge and X and will be a plus image with just equal to the final point that this be late. So this is our approximation for three months? Um, they no. There are two ways to calculate the area off the region bound by this ffx from a to B and obviously with respect to the X axis. So how what are those true is first way is that each sub mental basically we are taking too is off. Assuming our $7 so $7 we're zooming off like England. So if you consider this strip for this region are speak Q B. If you consider for this particular region, then there are always off taking rectangular strips off with H. So when is this R F you'll be this one, right? So this s one of the lake R s, Q B off taking a subliminable. But if it carefully notice that if you take some interval off this type R F u P then in this region R S Q p this artist of region or this region is left behind that means for this particular region. If we calculate if you take the rectangle step of this type well, living some off the area. And if we take the rectangles off this type through the through the through, our point through from A to B basically, we will be leaving behind some area. So let's see the idea found out by this method by taking the tango steps off this type. Let's say that this small isn't it is my taking the all you can see the smaller electing this by smaller rectangles. We mean that we are actually leaving behind some part of the area from each of direct, endless trips. So this is the submission off all the smaller rectangles like this Taken A soon buyer says small lesson No, The second day is if we take the signal a strip off this type like that, this e s you'll be this whole e s Q B right? This will E s Q B. So if you carefully notice here, we're taking some extra area. In the previous part, we were taking some less area. But here we're taking some extra area, right? So similarly, if we so some off all the areas off these rectangles, then we will be eventually taking some extra area with off which extra area than the area off the region, which we were intended. Look fine, right. So let's sit that this capitalism. This is the submission off all the larger rectangles. And since this is a little bit larger than the area, but we need to find out on the small lesson is a little bit smaller than this area, which we need to find our weaken come to a conclusion that the area is lying between small lesson on Capitol essence late. So this is the consideration till this point, so No. We will find out the formulas for each off this small lesson and capitalism. So small lesson was the some off all these smaller rectangles. So for this particular first, regardless strip it will be somebody like this. For this. It will be some back for some but like this. And let's say for the last rectangle it will be somewhat this. Thank So how we will find out the small listen so small. Listen, you can find out, but let us begin to own the first rectangular strip first. So that will be not this height but this point. It will be value of this function at this point that this excellent So the height will be effects, Not right. So this height will be FX not end of it. Obviously which we have assumed this at the eso The area off the sector analysts trip is if it's not times edge. Similarly, for the second step, it will be affects one time. Zetsche, Andi At the last, it will be f x n minus one time Such right? So no, we can write the formula for it's more Listen So that will be effects, not aims. Edge less FX one times ed less fx to times edge on dso on till the last rectangle A step that is f X, but x n minus one times each. And if you take s common, this will be affects north plus FX one and so on. Il F X in minus one, Right? No, this x not x one until excellent minus one were already calculated as a a play set. And finally it is xn minus one is a plus and minus one which big? So let's put these values here So this will be it's times f A f off plus f off a plus edge and so on l f off a plus and minus one h thank. And also since x n is equal toe a place an itch which is equal to our final point B, That means a press manage is equal, Toby. And if you try to calculate the value off edge this will be at is equal do being minus eight. What? And late. So that is equal to B minus a over. And so the final formula is somewhat off this type. Similarly, we can find out the yeah, pretty listen so captured. The lesson will be now how we will compute with capitalism. So for capitalism, we are considering, let's say for this trip, this one we're considering this site that does Sq and for this particular strip we have to consider FX are that means for each, like Bangla Strip, we will consider the higher X coordinate. So let's say for the first rectangle strip it will be the site will be affects one. And for this it will be effects do. And similarly for the last it will be effects in so we can write and we can draw this for the last one. So this will be off still the pill till here from the start, right? So this will be FX in signs It's which is the height Great. So our formula for capitalism will be f x one times edge less FX to times at and so on Still, FX end times each And if we take out its common so this will be affects one plus x two plus affects in and we will substitute the values off x one x two election. So this will be shapeless. Edge it. Let's do it on his accent will be a place, an itch they don't and similarly we can secure. The value of that is equal to B minus order. So we are able to compute the formed US for small lesson and capitalism like No, here is a consideration which we need to take into account that here we are taking and any call $7. And this with EJ is a financial right. But if we consider this with EJ will be very small. In that case, we'll assume that this is approaching 20 or we can say limit a limiting that is approaching 20 That means limiting value off at zero. Right, so it is approaching 20 So if it is approaching 20 that means our seven tells will be soon will be very large. So the quantity off seven tables will be very large. That means the end, which are the equal, which are the number of $7 that will be tending to infinity. So this is our approximation. So if we considered by that case will be close toe the actual area and in that case, and that is if we consider and approaching toe infinitely, which comes by the approximation at approaches to zero. Then the limiting value off s in, which is by the approximation at approaches to zero, is equal to limit a protester zeal A I was we need to find out off the called bicycle FX, bounded by the X axis from A to B. I mean, this is equal toe limit its approaches to zero capital s in. So if you assume an approaches to infinitely, that means the number of summit levels are so are very large. Andi says that David off If some interval is approximately zero, then we are actually close toe fine the actual area if we consider by this small lesson or if capital lesson So in the limiting case, we can find the actual area e by any off these formulas Small listen or capitalism they. So let's say the calculate for the sake of convenience from small is, um that means this form. So with cal appeared from this formula, so area will be equal toe in Diggle eight. Toby affects DX. This will be equal toe. Now this b minus, it is a constant, so it will move up and limit and is approaching to infinity one by N f off a plus F off a plus edge, less f off a plus sewage and so on l f off a plus and minus one h. So this is our final formula for finding the value off the area bounded by the co bicycle toe fx with the X axis from the points it to me. So this expression is defined as the definition off definite, integral as the limit off some. Thank you. 4. Fundamental Theorem of Calculus: Hello, guys. In this video, we'll be discussing the most important term off. Definite indeed. All that is the fundamental term off calculus. So let's start with this term. Let ethics be a continous function defined on or let's say, a continous function from it will be inclusive off book points, right? So a br two wins. So by this, we're assuming that ethics is indeed agreeable. That means in Diggle off if X exists they so if let's sit capital FX is the in Diggle or the anti derivative off FX, then what does the fundamental term off calculus see that in giggle from a Toby FX DX is equal to capital F off B minus capital F off a. So this is the fundamentally to him off. Get this they Which means we calculate the in Diggle off ethics as capital FX, right and then calculated the difference off FX at B and ethics at a So this will be our definite in Diegel. Right? So this is the fundamental term off calculus. Note that this F B minus story capital F B minus capital F A can build Dennis Capital F eggs and a on the corner of the square brackets. Be right. It does lower in the lower part. We'll write that right as a and upper part. Will I just be so This is another notation All we can write this as simply like this. So these are the other notations to represent this FFB capital effort being minus capital F off a right. So this is the fundamental term off calories. So this term is very useful because it gives us it gives us a matter off, calculating the definite and giggle more easily without calculating the limit off A some they. So this was it for this video. Join me in the next video that we discuss examples on the fundamental term off helpless. Thank you. 5. Fundamental Theorem of Calculus - Examples: in this video. Let's try our hands on some examples within. We will use the fundamental term off calculus. So let's start with our first example. Our first example is we have to well, wait and giggle from 2 to 3 X square NeeIix No sticking to the fundamental term off calcalist. First, we need to find the integral off this immigrant that is X squared. So let us see Represent that by capital ffx So capital F X is equal to in giggle off X squared. Right, So this is by powerful X Cube by three. Let's see So this is by this is the and did anyone off then or the integral off this interim right now by the fundamental term off calculus the integral the about integral can Britain US captain left at three minus capital F had to since the upper limit is three and the lower luminous too. So this means the answer is F off capital F of three minus capital f off to so this will be three Q by three The sea minus took you by three. Let's see not the sea will get cancelled from the sea and in the coming questions. Also, we will not be considering this concert off integration for the different integral by because in every evaluation from the difference off upper and the lower limit, the sea will have to be cancelled anyway. So we will not be considering this concert of integration from known. Right, So this will be simply three cube minus took you my three on three Cuba's 27. Do you miss it? So the answer is 19 my three. So this is our final answer. Like so, No, let's take another example. Our second example is I is equal to integral zero toe bye bye to got six dicks. So again, this is the indignant. So finally so festival the integral or the anti derivative off this whole sine X, we need to get it. So this will be cool signings. And now, uh, let's see, we were ignoring right as we as I explained you in the previous example. So capital F X is equal do signings. That means I will be equal toe capital f off the upper limit. It's bye bye to on the lower limit is zero so capital F bye bye to minus capital F zero so capital F off vibrato let us substitute so this will be signed off by by two minus Sign zero. So sign off by Baidu is one and sign of zero is zero. So the answer. This one, right? So this is our solution to the second example. Let's move on to the third example. Our turn example is I is equal to integral zero toe by before do six clinics plus X cube plus two right, So are indignant is, let's say f affects the indignant is who? Six clerics, less X cube. Let's do so. First of all, we need to calculate the anti derivative or the integral off dysfunction, so that will represent by capital affects. And as you see, there are three teams involved in the integral. That means we will apply this some rule to evaluate each other terms individually. That means capital F X is equal toe integral off six quakes DX plus in legal, off X Q B X plus integral off duty X. So this, too is a constant. So this move move all of the concept of the integration. So this will be do no integration off sex car wrecks This is a cool book annex since differential of panics is equal Deus ex convicts that that's why in legal of sex, Connexus Annex and in the middle of this part can be polluted by our room. So this will be extra the power three plus one that this for over three plus one that is four plus two is a constant and in the middle of Oneness Expo and we will be not considering the constant part like so our answer will be I will be equal to f off upper limit minus F capital F off 00 That is the lower limit. So first of all, for this park, then for this part and then put this part. So this will be a price off and by before minus and zero less one by four is a constant that will come up on the by before, through the power four minus zero to overpower this forest zero less rice off by my four minus zeal late. So the answer will be not price off. And by before this one, so one minus 10 0 is zero. So this will This will be a price of one that it's 22 Right. So this is a bit too. Let's now, this is by to the power for over four to the bar food and for the bar formerly black by. This is for the par five. So the answer is, from this time, it despite with the bar four over four to the power Fife. And this is my before monies erased by before. So this will be by back to. So this is less bye bye to So this is our final answer. So in this example, to calculate the anted elevator that is the F off capital FX, we applied this some route, and then we applied the already discussed fundamental term off calculus to achieve the answer. So these were some examples on the fundamental term of helpless. Thank you. 6. The Substitution Rule: in this video, let's see the substitution rule for definite intervals. So to evaluate in legal it Toby F off DX that is the definite integral from A to B my substitution rule. We need to follow four simple steps. So our first step is transformation step. What is this step? Let's sit There is a function in the integral. Let's say there is a function that is g affix with this president in the needle and it's very really or different station that is, deep remix is also present in the integral. Then, by assuming let's say any another, another function by is equal to g o fix we can transform. Or we should transform the above integral in this form by making this substitution right. So the differentiation off or function DX is president. That is deep remixes, President, we can make this transformation. So this is the transformation step. Our second step is no. Since we have changed the variable and the integral, the limits will also be changed. So our second step is change off limits. So we need to keep the integral in the new variable itself on. We have to change the limits accordingly. late, So this was our second step. Now, after changing the limits we need to. The third step is we need to find the integral. So, since we have transformed to the new variable, will be integrating the integration with respect to the new variable without mentioning the constant off integration since the constant off integration gets eliminated. So this one, our third step now the fourth and the final step is months. We have calculated the integral Let's say the Eagle was for capital ffx. Then we have to evaluate the in legal by the fundamental term off calculus. So the fourth step is evaluation, my fundamental to them off helpless. So this is our fourth step, you know. So let's say the eagle was FX, then by the fundamental term off calculus and the new limits, Let's say, a dash and dash, which we found out in the second step. So let's say that you knew new limits. The lower limit on the upper limit are a dash and dash, respectively. So the final answer will be capital F off be dash minus capital F off a rash. Remember, these are the new limits, which we found out in the second step. So this is the final answer, which we evaluated in the fourth step by the fundamental totem off calculus. So guys don't remember these four steps. These are the basic steps which we will be following in calculating the definite in Biggles by this substitution rule. Now, in the next video, we'll see some examples for this This rule. Thank you. 7. The Substitution Rule - Example: So in this video we will see some examples on evaluation off definite in Biggles by substitution rule. So our first example is we have the velvet an Eagle from zero bye bye to science critics Go six dx So viable. Apply the substitution rule here because we're able to see a function Who's David? That this koszics sine x d Wade Davis Koszics So we're able to see a function whose very where is present in the function itself they so our first step is the transformation step. So in the transformation step we assume a function whose derivative is also present in the funk in the Indian immigrant itself. So here we will assume why is equal to sign X since it's very, very diverse present in the function so differentiating both sides so de vie will be equal to Koszics aims differential affects that is DX, right? So the transformation off the integral will be. So this will be vice square and this whole will believe so the in eagle will be vice square leeway. So this individual gets changed to disintegrate. Okay, so this is the basic transformation. Now the second step is Jane's off limits No, you will be able to understand by this step vibe by we are introducing the step. Change off limits since we have a chance from from X to Y in the signal, the limits will also change since we assume via physical to sign X. So initial limits were in for for the variable X. Now it will be for the variable by right. So that means we have put James the limits since we will be keeping the integral in this variable. Okay, so at X equal to zero bicycle do sign zero. So that will be zero and at X equal to buy by do viable be equal toe sign off by by toe. That means it is one. So the new limits will be zero and one, respectively. Right. So we have performed the second step. That is the change off limits. No. The third step is to find the eagles find the integral. We have to calculate the integral which we find found out which we transform in the footsteps. So we have to calculate the indeed so in giggle off vice square D by This will be cool. Do invite you by three on we will be not considering the constant of integration since it will get eliminated indefinitely. Giggles. Right. So this is the integral. But you have calculated. So let's say this is capital F halfway they. And now the final and the fourth step is evaluation off the integral using fundamental que TEM off calculus. Great. So evolution using this step No. In the second step, we calculated the new limits as zero and one. That means the evaluation using fundamental term of calculus will be an act. Those image. So our final answer will be f off capital F off one minus capital f zeal. Know what this capital f off when that is like you by three. So capital F one will be one cube by three and f of zero will be zero cube by three. So this is one minus zero by three that is equal to one by three. So we were performed the four steps very carefully in order to find this answer for the definite individuals using the substitution. So this was our first example. Now let's see our second example. Our second example is we have the value it in eagle 0 to 1 x cooler X square plus months late. Now again as you can take a hint from the question itself that river devolved dysfunction X Square plus fun, which is to explore zero that is, two X is somewhat present in the numerator. So if we made a plan very bad, too, two x can come here on did delivered it will be present in the numerator. So that means this gives us a hint that we will apply this substitution room, right? So let's apply the four fundamental steps off the substitution rule. So the first step is, as you know, the transformation step. Now what is the transformation step? We have to assume a function would derivative is somewhat present and the indignant right, so that function here will be X squared, plus one. So we will assume why is it called Toe X Square plus one? This means let's differentiate both the sides. This means D Y is equal to two explosive ill. That is two weeks. I'm the differential effects that is DX. So devise equal to two Xbox or we can write as one by two. B y is equal to X dx, so this And this will help us in transforming this apparent giggle, which is 0 to 1 x over X squared, plus one to a new and giggle. Let's not talk about the limits in the step, right? So first of all, we just need to transform the Indiana. So in Diggle Xcor X squared plus one BX animal readiness knows in this X DX candidate and s one by toe diva. So this will be one by two DeLay and X squared plus one is by. So this will be one bye bye. Right. So this is our transformation step Now, The second step is changing the limits now how? Villa limits betweens that will be changed. Espera substitution, since we assume why is equal to X squared plus one. So why is equal to X squared? Plus one was our substitution. So at X equal to zero This why will be zero square pass one? That will be what? So I will be one and at X equal to one. This fight will be one square plus one that will be toe advisable toe. So this means these our these are our new limits that this one and two are our new upper upper lower and the A parliaments. Right. So this was our second step off changing the limits. Not 1/3 step that is finding the integral. So let us now find the integral which we changed in our transformation step. So this is this one. So one by two in Eagle One, by the way, So the new integral will be won by two in digital one by way, the way and as you know, integral often by way with respect to buy will be l unmoored by. So the answer the integral will be one by to Ellen more vie. And let's assume this as capital effort fight. Right. So this is our hard step. Let's be used to find the in deal So are integral capital f off. Why is one by two Ellen model and our fourth And the final step is evaluation using fundamental totem off helpless, which means we have to evaluate at the new limits for this particular individual that is capital F off new upper limiters do from the second step. So this is capital F to minus capital f the new lower limit that is one. So this is capital F off to minus cap. Left off. So which is one by two is common. And Ellen more to is to minus Ellen one and Ellen Win is zero. So the final answer is one by two. Ellen, no. So this is our final answer for the second example. So, guys, these were some examples for evaluation off the definite in eagles using this substitution room. I hope you understood this. Thank you. 8. Properties of Definite Integrals - Part 1: in this video, we are discussing properties off definite and giggles. So let's start with our first property. Ah, first property for definitely giggles is in legal zero to a ethics NeeIix is equal to in Tegel zero to a f. P. Diddy. This means we can change the variable in the indeed, all without affecting the limits. Our second property is integral. It will be f X dx is equal to negative off integral beato a ffx dicks which means that if we just reverse the limits, the integral sign will get reversed later. And the particular case for this rule can be when water the limits are saying. That means from eight to a India left off the ffx d X is equal to minus integral eight way if affects DX, that means in legal FX the X from eight to a will be zero. Since you can just transfer from the right side that integral. So the articles are saying that means the sentinel will be zero. So particular case is indeed all from a to a ffx DX that will be zero. So this is our second property now. Our third property is an eagle. It Toby FX. The X is equal toe integral eight to see ffx dx less integral See, Toby And for fix DX there, this sea is a point lying between a and B, right? So we can just split up this integral will point c and from then see will be right so we can do this splitting. So this is our third property now. Fourth property is integral it Toby fx. Vieques is equal to integral. It will be f off a plus B minus x the x So you can remember this as the placing exit the some off the limits and that is a plus B minus x so this will remain the same. So this is our fourth property Now. Fifth property is in giggle zero to a if affects BX is equal to integral off zero to a f off a minus x dx. So if you carefully also that this property is a special case off this property there, we assume is equal to zero on B is equal to a So if you play this property here, you will get the same result. Okay, so this is a special kiss off this at a 00 and be school to a. So these were five properties for property as the properties off definite in Vegas. No, Let's take one example out of these properties to explain you the properties and explain them how they are applied in action. So our example is we have to evaluate integral minus 1 to 2 more necks. NeeIix. Right. So first of all, let's focus on this function. This is more dicks. Marnix means absolutely. You fix for this function. Marnix is defined as X, then X is greater than equal to zero and minus X. Men access less than zero. And as you can see, the limits are from minus one people. That means from minus 1 to 0, it will be under negative values off X and from zero to every will be in the positive values of X. That means the definition off this morning will very for this particular in eagle from minus one football. So we need to apply a property here. So which feel which property will be a plane? As you can see, we discuss the property, this one that hard won. There we split, did the giggle to a point in between those limits status A and B. So we consider the point c in between to split the indigo Girls Now here Also, since the integral behaves, the immigrant behaves differently from minus 1 to 0 and from zero to level split it in two different parts from minus 1 to 0 and then zero do do you. So this will be our two parts. So we'll be applying that property that in Eagle it Toby FX be X is equal to integral A to C ffx DX plus integral See Toby if affects sticks and why we're splitting. Because FX, that is this Marnix behaves differently from minus 1 to 0 and then from zero to so here is it will do minus one. Si is equal to zero. That is a point in between and be off course this upper limit that doesn't kowtow. So let's apply the property. So the bride the property a single do minus one of minus one. Bees glued to go from minus one toe in eagle. So minus 1 to 2 mornings, DX is equal. Do minus 1 to 0. Now more decks from minus 1 to 0. That is X less than zero is minus X so this will be minus 60 X plus from zero to this will be positive X So this will be integral zero to x TX they so the integral will be the integral off minus X is minus X squared by two from and the relation of end up and we will be putting the limits from minus 1 to 0 plus in eagle off X from zero To do that is X square by two, and the limits will be from zero Do so this will be Oh, so we'll be evaluating to get the final answer. So let's take minus one radio comin from this from this park. So this will be zero square minus minus one square plus when my toe common. So this will be to square minus zero, which is zero. So evaluating further, this is minus one by two and minus one square is one. So this will be one, and this will be minus one on this will be four by two. So this will be minus one times minus minus one. So the answer will be one plus four by two. That is equal to five by two. So this was an example on the application off a property off. Definite indicates 9. Properties of Definite Integrals - Part 2: in this video will continue with some more properties off definite in giggles. So our six property is in Kagel zero To do it ffx DX is equal to in eagle zero to a FX DX plus in eagle zero toe a f off minus X dx. So this is our six property. The next property is an eagle zero to do a and fix DX is equal to price off zero toe a indeed, a zero to a fixed dicks. So this is true if f u N minus X is equal to ffx and it is equal to zero if f off away minus X is equal toe minus off FX. So this can be proved by assuming that if the way minus X is equal to FX, then we put replace this way ffx in the girls it away f x dx. So this will become the ice off the roadway. Ffx dicks right on if it is negative, off affects this to women in sex. So this will become minus and eagle zero to a F f f x dx. So this will become zero, which is the desert in here. Right? So this is our seven property, our final property and the eight property is in Diggle minus a to a ffx DX. This is equal to rice off zero toe, a integral affects DX. If FX is an even function now, what is even function by even function, We mean that f off minus X is equal. Do f or fix. So if you're function ffx satisfied this property than it is said to be an even function. So for an even function minus eight to a niggle fx T X is equal toe myself in equals zero to a FX teas other ways. If for our function, this thing is equal to zero that this if FX is an award function No, again, what is an award function or function is a function which satisfies great idea something like this that f minus X is equal to minus off FX like so these are the remaining properties off Definite, indeed, bliss. Now let's take an example from these properties. So our example is we have to integrate from minus pi before toe by before sine X and his Deeks. So we have to integrate sine x with respect to x from minus by by four Dubai before, right? So anything off a property of its weaken a play. So from these properties, which property can you black? So if you closely observe that here, sign ICS if it closely observed that sign off minus X If you think off, it is equal to minus sine x They this part, you know. So if this such as the thing that sign off minus X is equal to minus sine X And from here I mentioned that if FX is ik is an art function, then f off minus X is equal to minus affects. That means sine X is and heart function. Okay, So if Sine X is an art function on, we had a property that in eagle minus eight to a FX D X, is equal to zero. If FX is an odd function, so the limits are something like minus eight to a. There is equal book by before so by the property that if FX is an art function and we have to evaluate in Diggle minus 80 a that fixed DX, then this will be equal to zero. So by this property we can see that integral minus pi before toe by before sign off X in giggle of this. No, this is an art function that means from minus pi before to buy before in Eagle of Sinus will be nothing but zero. So this is This was an example which we look from our just now introduced properties In the next letter, we will continue with some more examples on these properties off definite indignance. So stay tuned for those videos. Thank you. 10. Properties of Definite Integrals - Examples: The motive off this video is to lay a solid foundation for the uses off properties off definite integral. So let's start straightaway without first example. So our first example is we have to evaluate integral want to do X multiplied by three minus X whole square. So as you can see the eagle off this three minutes x whole square multiplied by X. If you multiply all the terms, this will be a little plenty toe value it right. So let's try to focus on some of the properties if we can apply one of our properties We have just discussed here to make this integration little simple. So as you can, also, that some of the limits that is one plus two is equal to three. So here, f off x, the property one of our property is integral. A Toby ffx DX gangrenous Integral it Toby F off a plus B minus x dx They so if here three months six comes and here x, it becomes X, then it will be fairly easy to help it late. So if we apply this property that integral it will be affects the excess a wrinkle equal to integral a Toby F A plus B minus x dx. Then I think we can make this question look a little simple. So let's first apply the property. So our properties 1 to 2 No, this X multiplied by three minutes. X square, be X. This will be equal. Do want to do now. A plus B minus X is one plus two minus X that is three minutes X So this X becomes three minus six and this three minus this excess three money, sex So three minus three minus x is X itself. So this is X square. No, you You can see that if we apply this property, this question becomes little simple as we just have no totems needed. Why? Because we can now straightly straightforwardly multiply this X square with three and the next square. With this extra, this will be integral. Who want to to three x square minus excuse. So this is no fairly simple. So integral Oneto three x minus X cubed. So this is in legal. Want to do three x bx minus in giggle? Want to do X Q. The X now we can easily apply the powerful in order to evaluate the answer. So this will be three will come out of the integration on integral off excess X square by two. So this will be X squared by two on the limits will be applicable on this from 1 to 2 minus . This will be extra. The power four by four on the limits will be from 1 to 2. So the answer will be This will be this one by the will come up as a constant on and it will be to square minus one squared that is one minus one by four comes as a constant on. This is to do the power four minus one to the part for that is equal to one. So this is a three by two times four minus one is three minus one by four. This is 16 minus one that is 15. So this is equal to nine by two minus 15 by four. And let's take a common denominator. So this will be a in by four minus 15 by four. That is three before. So the answer is three before. So you have seen that we applied this property in order to make this integration Look fairly simple. Otherwise, we had to multiply all the items inside this that would have been more hard work in evaluating the needle. So this property had. There's a lot in evaluation evaluating the integral. Now let's take another example. So our second example is evaluate integral from minus pi before to buy before signed Square X, the X. So we have to alert the giggle from minus by before toe by wherefore for this science works . No, we have seen a property off this kind as something like this when we were using a property for the integral from minus eight to a FX takes. So there were two cases that either if the back property is applicable, other if the function is an even function or if the function is an or function right. So let's first try to find out where that this Science Square X is a even function or an order function on and off these. So let's first find out. So let's say, if X is equal, do science works made so let's calculate f off minus X for this. So this will be a sign off minus X hold square and what is sign off minus X sign off minus X is minus sine x and this is holed square No, minus sine X hold square is this minus one square is again positive one. So this will come as science critics so you can see that f off minus X is equal to F off X . So if such is the case, this means that ffx is an even function lady. So if FX is an even function and by the property which we have discussed that if in Diggle we have to evaluate from minus eight away effects DX then if it is an even function, then it will be called to price off integral from zero to a FX ticks. So let's apply that property. So this means that minus integral from minus pi before to buy before off Science clerics, this will be equal to the ice off integral from zero to a s by before sine squared Xbox. So now we have just find this indeed all and multiplied by two in order to find the answer . So now how will find this in Eagle? Science collects from zero to buy before now, As you might be aware off the property that in big cost two x is equal toe one minus two science critics. This means that science could expect in right as one minus cost two x by right. And now we can apply this identity to transform this science collects into this form and vibe. You're transforming this. We're just using this identity because we know the integral off this part, which is quite easy in legal off course two x It's quite easy to compute, so that means that rather than integrating science Culex we can integrate this fairly easy . So let's apply this over here. So this means it's the integral eyes equal to twice off, zero to buy before one minus cost two x by two. Right on This took and began slowed by this to since both are constants and equal to two right, So this means we have to know indicate from zero to buy before one minus cost weeks late. So in Tegel, zero to buy before one minus goes to x. So this is we can do by the some room. We can split out these totems, so this will be zero to buy before in Diggle off one DX minus in legal off zero Dubai before course two x the X right. So in the middle of Oneness X So we have to play limits on X from zero Dubai before and in legal off course two x is signed two x divided by two as we multiplied by two in different station. So here we will be dividing. But since indeed off course X is sine X by a So that's why we are integrating this cost two x by signed to expect toe. So this is our answer for this time and we will be evaluating the limits I from 00 Dubai before. So let's apply the limits. So it will be by before minus zero. Ana, this will be one. By the welcome. I come out as a constant. So this will be signed off by wherefore which is signed off by by two minus Sign off to time zero that this sign zero So this is equal toe by before minus one. By do now signed by by two is one and sign 00 So this will be fun only so the answer will be by before minus one by two So this is our final answer for the second problem. No, let's focus on our for example. So our third example is we have to evaluate Integral zero to buy my to sign ICS Well, later signings less classics late. Now let's try to think off a property which you can apply here in order to make this integration look fairly simple. So as you guys might be knowing that sign off Bye bye. Minus X This is equal to us off by by cost cortex, right. Similarly goes off by Baidu minus X is again equal. Do it is vice versa, right? It is equal to sign it. Sure, if he used the property somehow like zero toe Indeed ALS zero to a FX TX is equal toe in eagle zero to a f off a minus x dx. Then I think this stone will become classics. This will become signed core sex and this will become cynics. Right? So in that is the sum off sine X plus score six will again be the same since the terms huh will be reversed. But derision will be same. So the denominator part will remain the same if you apply this property on Lee, the numerator will be changed that will be called to co sign X. And if we add those two totems, then I guess these ignominy functions and get easily eliminated. So I guess it's a quite helpful property in order to evaluate this integral. So let's see that this property in action by the property zero to a giggle affects D X is equal to indeed all zero a f off a minus x dx. Our big girl was, Let's say I was equal to in legal from zero to buy by to sign X over sine X plus co sign X . So let's say we assume this as I and we apply this property so again, weaken right I as we substitute from here to the right inside. So this will be zero to a So this is sign off a minus X. What is a despite but so sign off by by two minus X over. Sign off by by two minus X. Let's go sign off by by Dome in Essex and this is DX. So as we have discussed that sign off by by two minus X isco cynics and course off my bedroom. NSX is signings, so let's apply those identities. So this will be close X over core sex, less signings late. So if you see that these dumps if we let's say this is one on let's say this is too So if you know carefully see that if we add these totems, then in the numerator it will be signed explode score sine x on in the denominator. Also, it will be sine X plus course I next so that the lead book cancellation off the numerator and denominator. So this is a really helpful property which we apply it just now. So let us go on further toe, add up these doing questions, so let us add both went and toe. So if we add up both one and two, this will become I plus I is equal to in giggle zero to buy Baidu, sign ICS over sine X plus 06 the x plus integral from zero to buy by two Koszics over course express signings they millets Look does this indignant and disintegrate together so by the symbol if we have would have different she made this divided into totems. Then we could weaken also join these totems, right? So it is equal to zero to buy by two on the numerator part iss sine x plus koszics older This is sign expressed Koszics And now you see that this whole sine X plus Koszics gets canceled with this sign Explosive sex. So what it remains is only one And in the left side this IBIs I is equal to the way right So to I now becomes in eagle from zero to buy by 21 day X And this is now very, very simple. This is integral of oneness X So final integral comes out to be x and the limits are from zero to buy back. So if we apply the limits this will be bye bye to minus zero which is equal to buy way too now Since the way was equal to vibrato on what? We have to evolve it. We have to value it I which we assume from here So we have to value it So I will be bye bye two divided by two that will be equal to buy back for So this is our answer. So the turn Example these words of examples to make you comfortable with the properties off definite Integris. You can find these properties in the reference section off the course for your easy difference. Thank you. 11. Miscellaneous Examples - 1: You know, we have discussed all the concepts of definitely and giggles. Now it's time to try our hands on some mixed examples for these concepts in order to make you comfortable in selecting the appropriate concept. So let's start without first example. Our first example is we have to evaluate from 1 to 2 integral off by square less one by Vice Square. No, this can be read it and better as Vice Square, Let's invite to the power minus two. No, Here we will apply the some rule off indication That means we can split out both the dumps on evaluated individually. So this will be 1 to 2 integral Vice square Divi plus in Teagle Oneto by to the power minus two giver so that by the power rule this will be Vike you that is right to the power to plus one by two plus one that is three on the limits will be well worded from one toe and here also we can apply deep power. So that will be right with the power minus to plus one over minus to plus one. And here also we will apply the limits from 1 to 2 so this will be no. Like you buy three when my trick will come out. So this will be due to the power three minus one. Bless. Now, this will be by to the power minus one over minus one. So this is minus one. Bye bye. Late, That's what This is minus one. Bye bye on the limits will be from one toe. They So this is one by 38 minus one is seven minus one by way. So it will be won by two minus one weapon that this work. So this is seven by three, seven by three Minus. This is three by two late. This is when they don't. Sorry, this is one by two minus one, which is equal to minus one wayto So this is minus 1 May toe. So this is turned by three. Plus one by two. Let's take the common denominator, which is Elsom off the student at the six. So this will be 14 plus three. So the answer will be 17 by six. So here you see an example. We applied this some rule and the important power rule in order to evaluate the integral No , let's take second example. So our second example is integrate 1 to 2 Ow to the power five minus w plus three over the Blue Square. And obviously it will be need a blue since the integration is with us with respect to w So how we will devalue it So as you see, if you split out all the items one like this other like this on the tournament like this, it will be quite better in order Do indeed it. Right. So let's spit out all the freedoms. So this will be integral. Oneto good. A blue, the blue to the par five over the View Square DW And also we're applying with some rule. So we're evaluating the in giggles individually. So this is minus not a blue over W square is one by W So in eagle 1 to 2 dw one by one by W w bless integral off three by the blue square They so this is to w Q so too will come out as the factor And it was here, went to to so to will come out as a constant. So this is 12 to the blue que minus in eagle off one by the blue will be Ellen More w So this is Ellen. More w Onda limits are from I want to to. Plus, in legal now, three bite abuse square three will come out off the integration and this candidate and us double the power minus two on the limits are from 1 to 2. So now, by the power rule, this dumb will be and giggle off. The blucher will be doubled up. Our 4/4 limits will be from 1 to 2. And this is Ellen minus Ellen. More to so more to is to so minus Ellen to minus Ellen one plus it also, we will apply the power rule. So this will be the brutal power minus one digest one by the rutabaga minus one over minus one. So that is minus one YW and the limits are applicable from 1 to 2. So this will be do by four, since four will come out when before will come out as a constant. And this will be true to the power for minus one. So the limits. After putting the limited, we were told that the power for minus one and Ellen one is As you know, log off when with any base is zero. So this will be zero. So from here we will get minus Ellen. So this will be minus Ellen. And it is minus three in one by w Right, So minus one will come up. So this will be minus three. No, we will be putting limits on when by W from one. Right. So this will be No. This is one by two. And what about four is 16 16 minus one is 15 minus Ellen to minus three one by two minus one by one. That this one. So this is equal do 15 by do on. Let's group the constant part together. So this is one by two. Mannesmann that is minus one by two multiplied when ministry That is plus three by two. Minus. Ellen too. And this is 18 by do, which is nothing but nine. So the answer comes as nine minus Ellen. Do so one thing. I just want to tell you that by starting with this question, you could think off like this is a vessel function, and you could start with proceeding by while driving this right so you could you could you could be confused that maybe we need to apply some approach off the national functions. But if we were just splitting the comes like this, it is quite easy. Actually, we will load the eagle just by using the power. So now let's focus on the for example. So are, for example, is a little tricky. No, we have developed from minus 1 to 5, one plus w multiplied by two W plus the blue square on this whole list to the power five. And this is a D. W. So this is looking a little complex in order to evaluate. So let's think off a rule which we can apply here. So if you carefully observe this pipe that is to W plus w square. So if you see that differential off, this could've blue plus w scalable be this will be to, and the defense selloff through the blue will be to on this will be to W. So this will be to test out of you, which is somewhat present in the question itself. If he multiply this by right, that means we can evil. We can evaluate this integral by using the substitution Okay, So let us going forward to valued using that group. So let's say U is equal to any variable do w plus W square. Since they were two of this is president, It's somewhat present without adjusting the constant. So that means we will assume this s another fun function. Let's say you anarchists, So let us differentiate on both sides. So this will be Do you Is it going to do? Plus, do w indeed a blue on If you take two as a common, this will be two times one plus w dw, which means one by to do is equal to one plus w dw So this thing can be replaced by you on this one plus w time state of you can be replaced by one by to do so. Let's apply the transformation here to change this integral now, since the in giggle is getting changed by the transformation rule. So this was our first step in the transformation rule in the substitution rule that we have transformed the integral from first variable from a variable to another variable in order to make this integration simple. So this was our transformation step so as you remember we mentioned four steps using this procedure, right? So what was our second step? Second step was change the limits. Since the variable has no changed from the blue to you, that limits will also be teens, Which means I Now let's let us evaluate the new limits. So at abuse equal to minus one use, it will do twice off, minus one less minus one whole square. So this is minus two plus one, so that is equal to minus one. So at the bill equal to minus one, U is equal to minus one on at the blue, equal to the upper limit, that is five. You will be equal. Do good names. Five plus five square. So this is N plus 25 that is equal to 35. No, we have calculated the new limits. So our third step was we have to find the integral. So finding the integral now what was are integral between transform from the blue to you so integral will be that this part will be one plus w multiple times did a blue will be won by to do and this will be you to the bar five So our new niggle will be in eagle one by two, issued to the par five new. Right. So this is our new needle and the new limits will be minus one, 2 35 since we that's really calculated in the previous step. So this will be our immune giggle on. We have to find a needle. So let's say f off you as the in giggle off this part. So let us calculate the eagle first. So this will be won by to you to the bar five when by the welcome more since it was a constant, so it will come out of the integration. Now let us evaluate the integral for this part. So this will be evaluated using our lonely. So this will be U to the power six by six. Right? Since we're just evaluating the integral, we are not putting the limits. So this is your to the power six by but so we have calculated capital f off you, which is the anti derivative order and eagle off this part. Now the fourth and the final part is evaluation off the integral using fundament e to them off Galaxies. So what was the fundamental term of calculus. It was the The eagle is equal to the difference off the anted elevated. That is f off you cap the capital f a few here. So the difference off this function at the upper limit and the Lord Yamin and you see that the limits are that are off are the new limits since we are expressing in the form off you so f off 35 which is equal to 35 to the power 6/12 minus F off minus one. So minus one to the power six is minus one. Sorry, It is a positive one. And then the power is even so minus one to the power. Even power is one. So this will be eaten one by. So the answer will be It would be five to the power six minus one Where? So this will be our answer 12. Miscellaneous Examples - 2: So in this video, let's continue with some more examples on different in giggles. So our fourth example is given that integral due to seven off a function f x The X This is given us 20 Onda integral 4 to 7 FX Dicks. This is given us 30. We have to evaluate and giggle do before FX DX. So let's start with the solution. So given in the questions is this on this? So from 2 to 7, the individual is given and from four per seven it is also given, and we have to compute from 2 to 4 late. So if you remember, there is a property that integral from it. Toby F X DX is equal to in tingle from a to C f x dx plus in Diggle from See Toby F X T X. There the sea lies between A and B David. So if we assume is equal to two, Andi B is equal to seven on a midpoint. C is equal before because in the question from 2 to 7, the signal is going on. There is a point which we can see in between that is equal before, That's why would it. Look, middle point c is equal before then. If you try to put be given data in this known property, then it would be like integral from 2 to 7. And for fix, DX is equal to integral from 2 to 4 f x dx plus in legal from 4 to 7 fixed eakes. Now this park is known from the question that this is equal to 20. So let's replace this. What can be and also in Diggle from 4 to 7 is known. So, which is this part on this part is equally toe 13. So this means don't be is equal toe integral you don't do before affects the explicit acting. So if the starting goes there, this implies that in giggle do before fix. DX is equal to 20 minus 13 that is equal to seven. And this is what we need to compute in the question. Right? So we are successfully ableto evaluated in Eagle 2 to 4 f x dx by using this property. So this is our final answer. No, let's move on to our fifth example. Sure, 50 example is evaluate in giggle from minus by to buy X dx older a square cost correx less the square Science critics Not this question. Maybe looking Give complex to you, but I'll suggest you a hint. When they were, the limits in the function are of the form minus eight to it. There must be a catching the question that the we can apply the property as we discuss previously that integral from minus 82 a f x dx. Either it can be twice off integral zero to a FX TX or it can be zeal, not this case becomes true. If this is an even function, that is effort. FX is an even function, and this case becomes true if FX is an art function. So whenever I try to suggest you, whenever there are limits off the phone minus eight way, the straight to find out whether FX is an even function or in or function. So, first of all, try for this thing, right? So let us try to compute over this complex looking function is exactly either even function or not function. Let's track first tryto do that, So if we suppose FX is equal to X over a square course, critics bless be square Saints critics so in F If f of X is an even function, this F X is equal to f off minus X and if it is an odd function, this ffx will be minus off F minus six late. So let us first try to find out f minus X. But this F minus x that lucky. So let's a place expert minus x so this will be minus X over a square square minus six Now also for minus X is koszics itself. So this will be cost square exit cells unless the square signed minus six. Hold square right now Sign off minus X is minus sine x on this minus annex gets squared again. So minus Cenex whole square this will be equal to science critics. So our final F off minus X will become as minus X over a square cost. Kerekes, Let's be square science quicks. Now you can see that this thing is negative of this. That means f off. My sex is minus off FX and from here you can see if f off X is minus off f minus X or F minus x here minus. If it goes there than F minus X if it is minus off FX with this true, in our case, that means the function F X is an old function. So if the function ever fix this is an art function. This means that in eagle from minus by Dubai, the FX what was FX? It was Explorer a square cost critics Let's be square science critics The integration off hold This complex looking function will be equal to zero. So this is our answer. So, guys, these were some miscellaneous examples which were considering the concept discussed in the definite and ego. Thank you.