For more information, please see full course syllabus of AP Calculus AB

For more information, please see full course syllabus of AP Calculus AB

### The Definite Integral

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- The Definite Integral 0:08
- Definition to Find the Area of a Curve
- Definition of the Definite Integral
- Symbol for Definite Integral
- Regions Below the x-axis
- Associating Definite Integral to a Function
- Integrable Function
- Evaluating the Definite Integral 29:26
- Evaluating the Definite Integral
- Properties of the Definite Integral 35:24
- Properties of the Definite Integral

### AP Calculus AB Online Prep Course

### Transcription: The Definite Integral

*Hello, welcome back to www.educator.com, and welcome back to AP Calculus.*0000

*Today, we are going to talk about the definite integral, very important.*0004

*In the last lesson, we used the following definition to find the area under the curve.*0010

*Let me see, what should I do today?*0015

*In the last lesson, we use the following definition to find the area under a curve, f(x).*0024

*f(x) is the function, the curve of the function but we just say curve f(x), that is fine, on the interval ab.*0055

*We said that the area was equal to the limit as n goes to infinity.*0065

*N is the number of approximating rectangles, we take the rectangles thinner and thinner,*0070

*of the f(x sub 1) Δx, + f(x sub 2) to Δx, so on and so forth, f(x sub n) Δx.*0076

*If we have 15 rectangles, we have 15 terms in the sum.*0095

*Δx was the width of the particular rectangle, we just added them up.*0101

*I will go ahead and express this as a = the limit as n goes to infinity.*0109

*We have the summation notation, the shorthand for this.*0114

*When we picked some index i, which runs from 1 to n, whatever that happens to be.*0119

*F(x sub i) × Δx, here the Δx is equal to the b – a/ n.*0125

*It is the right endpoint of our domain, the left endpoint of our domain of our interval divided by the number of rectangles that we wanted.*0137

*x sub i is actually in the little sub interval x sub i - 1 x sub i.*0148

*All this means is that when we break up the interval, let us say we have x sub 3, and we have x sub 4, *0160

*something like that, and we have a over here, b over here.*0170

*In this interval, our x sub i is somewhere in here.*0174

*Sometimes, we pick the left endpoint, sometimes we pick the right endpoint.*0179

*Sometimes, we pick the midpoint, and that it can also be any other point in between.*0183

*That is all that that means, this part right here.*0187

*Let me go ahead and actually draw a little picture of that.*0192

*We have our axis, we had a curve, we had a, we had b.*0198

*A is the one that we called x0.*0205

*Let us say we had 1, 2, 3, 4, this would be our x sub 1, this would be our x sub 2, x sub 3, x sub 4,*0208

*and our x sub 5 is going to be our b.*0221

*This just means somewhere in there.*0226

*Our x sub 4 is going to be somewhere in that little sub interval, that is all that means.*0230

*We are going to define the definite integral.*0241

*We now define the definite integral.*0251

*I will go ahead and put this in red.*0263

*You know what, I think I want to use red.*0273

*I think to prefer to just go back to black, how is that?*0278

*We will say let f(x) be defined on the closed interval ab, divide ab into n sub intervals.*0284

*Everything is exactly as we did before, each of length Δx which is equal to b – a/ n.*0311

*We will let x sub 0 = a, then we have x1, x2, and so forth.*0328

*We will let x sub n = b.*0338

*The x sub 0 is the left endpoint, the x sub n is the right endpoint.*0344

*Then, the integral a to b, f(x) dx, the definite integral of the function f from a to b is equal to the limit that we had.*0349

*It is equal to the limit as n goes to infinity of the sum i = 1 to n of f(x sub i) Δx.*0369

*Again, where x sub i is in the sub interval, x sub i – 1, or the x sub i is somewhere in the interval x sub i - 1 x sub i.*0384

*In other words, x sub 3 is going to be somewhere between x sub 2 and x sub 3.*0411

*That is all that means.*0420

*If this limit exists, if the limit on the right exists, in other words,*0422

*we go through the process of finding the Δx which is the b – a/ n, we find the f(x sub i) based on the x sub i.*0438

*We form this thing, we form the sum of this thing, and then what we get is going to be some function of n, and then we take n to infinity.*0446

*If we end up getting a finite limit, if the limit exists, we say that f(x) is integrable.*0454

*This, the limit that we get is that definite integral of f(x).*0474

*If not, if the limit does not exist, then not.*0485

*If not, then not integrable.*0489

* This is the definition of the definite integral.*0496

*We started up with an area, we use this limit to find an area.*0500

*Now we are actually using this definition of area.*0506

*No longer is area, we are actually defining it as something that has to do with the function itself.*0511

*Something called the integral of that function.*0519

*This thing, a to b of f(x) dx is a symbol for the entire process that we go through, *0527

*entire process of summation + the limit as n goes to infinity.*0548

*This is a symbol for the entire process of forming the function, taking the sum of the function, then taking the limit.*0564

*Let us label a couple of things here, say what they are.*0573

*This thing, the f(x), this is called the integrand.*0581

*This right here is called the lower limit of the integral.*0593

*This right here, analogously, is called the upper limit of the integral.*0599

*When we read this, we always say the integral of f(x) from a to b, not from b to a, from bottom to top.*0606

*This thing, this reminds us of the independent variable.*0615

*If the independent variable of the function is x, this has to be dx.*0628

*If the independent variable of the function is r, this has to be dr.*0632

*Reminds us of the independent variable and is the differential version of the Δx.*0636

*This symbol, it is in the shape of an elongated s to remind us *0675

*that integration is just a long summation problem.*0697

*As it turns out in mathematics where the only thing that you can never ever do is add two numbers,*0721

*or 3 numbers, or in the case of calculus an infinite number of numbers.*0725

*That is what this says, this is reminding us that all we are doing,*0730

*when we are taking the definite integral of something is we are forming a big sum, a hundred numbers, 2000 numbers.*0734

*Or when we pass to the integral from the limit, when we take the limit and we passed to something else, in this case, the integral an infinite sum.*0743

*That is what is there to do, it is their to remind us that all we are doing is we are adding a lot of numbers together.*0751

*The definite integral is a number, if it exists.*0761

*If f(x), if the function that we happen to be dealing with is greater than 0 for all x in the particular interval ab,*0784

*in other words, greater than 0, it means above the x axis.*0798

*Then, this number just happens, that is the important thing, it just happens to be the area under a curve for f(x).*0804

*We started off by dealing with curves and we use this limit that we defined, the limits of the sum of such and such.*0829

*It turns out this integral is the deeper concept, the integral is a number.*0838

*It is a number that is associated with a function over a certain integral.*0845

*We define that number by going through this process of doing the limit.*0849

*It so happens that, if the function is greater than 0, the area under the curve happens to equal the integral.*0853

*We are not defining the integral in terms of area.*0863

*What we are actually doing is we introduced area first *0866

*to sort of give some physical conceptual motivation, for this thing that we are trying to do.*0868

*As opposed to just introducing that this is the integral.*0875

*Here, you form the sum, you take this limit and that is that.*0877

*That is very abstract, we start with the area.*0881

*We are defining this, we use that limit to define the integral.*0884

*It is the integral that is the deeper concept.*0889

*It just so happens that it happens to match the area under the curve.*0894

*It is the integral that is the deeper concept.*0898

*It can stand alone, it does not need to be associated with area.*0900

*As a mathematical tool that happens to give us the area, that is what is happening.*0907

*Let us talk about that.*0915

*We just said that if f(x) is bigger than 0 happens to be equal to the area.*0925

*But if we have something like this, let us go ahead and go this way.*0930

*Something like that, let us say this is a and let us say this is b.*0939

*Let us just call this a’ and let us call this point b’.*0953

*For regions from a to a’, the graph of the function is above the axis.*0964

*If I take the integral from here to here, I’m going to get an area.*0973

*The same thing here, from b’ to b, the function is above the x axis, it is positive.*0977

*Therefore, if I take the integral, if I take that limit, I’m going to get a and area.*0983

*What happens though if the function is actually below?*0989

*Here, the area is positive, here, the area is positive.*0992

*The approximating rectangles are just, you are taking some Δx and you are essentially taking some f(x).*0997

*F(x) is negative, what you are going to end up with is a negative area.*1007

*For the regions below the x axis, the area, I will go ahead and put it in quotes, is negative.*1013

*Because this f(x sub i) dx, because these f(x sub i), these are negative.*1039

*They are below the x axis.*1053

*F(x) here is down here, -16, -14, -307, whatever it is.*1054

*These are negative.*1061

*When you take the integral from a to b f(x) dx from here, all the way to here, which involves this and this, this is the definite integral.*1070

*But it does not give you the area under the curve, it is a number.*1091

*What it does is it gives you sort of a net area.*1096

*It is a net area, if you will.*1102

*Let us say this area here was 10 and let us say this area under here was 30.*1107

*Let us say this area under here was -50.*1113

*The integral from a to b of this function is going to be 10 - 50 + 30.*1121

*I mean, technically, we can go ahead and take the absolute value, if we were asked what is the area under the curve?*1131

*We could take 10 + the absolute value is -50 which is 50, 60, and 70, 80, 90.*1138

*Yes, the total area, in terms of actual physical area is 90 but the integral is not 90.*1144

*The integral is 10 - 50 which is -40 + 30 – 10.*1152

*The integral is a number, it can be associated with an area if the function as above the x axis.*1158

*But it does not have to be associated with an area.*1164

*It just so happens to be the same sort of tool that we can use to do so.*1166

*Integral, separate mathematical object altogether, has nothing to do with area.*1170

*That is the important part.*1177

*The definite integral is a deeper concept, I should say it is the deeper concept.*1184

*It is a number which we can associate with a given function on a specified interval.*1208

*In other words, if I gave you some f(x) and if I gave you some interval ab,*1236

*with that function on that interval, I can associate some number.*1246

*In other words, I can map a function on an interval to a number.*1254

*That is what this is, mathematically, that is what is happening here.*1259

*Given a function and the interval, I can do something to it.*1264

*In other words, I can integrate it.*1267

*What I get back, what I spit out, after I turn my crank and do whatever it is that I'm doing to this function, I get some number.*1271

*For functions greater than or equal to 0, we can then associate area with this number, the integral.*1281

*The integral comes first, the number comes first, the area afterward.*1309

*We just introduced it in a reverse way.*1317

*Note the definite integral, if integral of a function, I guess the real morale that I’m trying to push here,*1328

*the definite integral of a function does not have to have a physical association.*1345

*The definite integral exists as a mathematical object.*1367

*It exists as a mathematical object, independent of physical reality.*1382

*The mathematics that we used to describe the physical world, it is not there to describe the physical world.*1396

*The mathematics exists, it is a tool.*1405

*Whether we discovered that we can associate this math, *1410

*we can use it to describe the physical world but it does not come from the physical world.*1414

*In other words, the existence of the physical world does not give birth to mathematics.*1421

*Mathematics exist independently as mathematical objects.*1426

*It just so happens that physical reality happens to fall in line with mathematical description.*1431

*That is what is going.*1436

*The integral exists, it happens to be associated with an area, if we needed to.*1439

*As we will see later on in future lessons, it is going to be associated with more than just area.*1443

*It is going to be associated with volumes and all kinds of things, which is why calculus is so incredibly powerful.*1448

*Let us say a little bit more.*1461

*We chose Δx = b – a/ n, in other words, I should that is, we made Δx the same width for every integral.*1464

*Δx the same for every approximating rectangle.*1489

*They do not have to be equal, they do not have to be the same.*1510

*Always approximating rectangles, we made them that way, of uniform width,*1522

*simply to make our lives easier so that we can actually do some work, do some math.*1526

*They do not have to be the same, each one can be different.*1533

*You actually end up getting the same answer.*1535

*They do not have to be the same, it simply makes things easier to handle.*1539

*Yes, that is the only reason.*1543

*It simply makes things easier to handle.*1546

*Also, we said that x sub i is in x sub i -1 to x sub i.*1560

*By choosing right endpoints, in other words by choosing x sub i as the right endpoint of that particular sub interval, that is x sub i = x sub i,*1580

*again, we just make things easier and we make things more uniform, more consistent.*1611

*It is not so haphazard, not without just taking random points.*1622

*We are being nice and systematic.*1625

*In any sub interval, we take the right endpoint.*1626

*That is it, nice and systematic, keep things orderly.*1629

*Again, we just make things easier.*1630

*Or I should say, not every function is integrable.*1640

*Just because a function exists, it does not mean that we can actually integrate it.*1644

*In other words, when we run through the process of finding it, taking the summation, taking the limit, the limit may not exist.*1650

*Not every function is integrable.*1656

*Our theorem, if f(x) is continuous on the closed interval ab, if it is continuous,*1667

*or if f(x) contains at most a finite number of discontinuities, *1698

*the operative word being finite, then f(x) is integrable.*1723

*We have a criteria now to decide whether a function is integrable or not.*1734

*If the function is continuous on its domain, on its domain of definition, on its interval that we have chosen.*1739

*If it is continuous, it is integrable, I can integrate it.*1744

*I can find the number associated with it.*1749

*Or if that function on its sub interval, the ab has a finite number of discontinuities, I can integrate it, it is integrable.*1750

*For now, we will evaluate the integrals, the definite integrals.*1768

*We will evaluate the definite integral using the definition.*1787

*In other words, we will use the summation limit.*1798

*We will use the definition to evaluate these things.*1799

*It is going to be a little tedious, but that is fine.*1802

*We want to get some sense of what is actually involved here, using the definition.*1805

*Later of course, the same way that we did with differentiation.*1813

*Remember, we have the differentiation, we formed the quotient f(x) + h – f(x) divided by h.*1816

*We simplified that, then we took the limit, that gave us the derivative.*1822

*That is the definition of the derivative.*1825

*Then, we came up with my simple quick ways of finding the derivative.*1827

*We will do the same here.*1830

*We are going to use the definition first and get our feet wet with that, just to get a sense of what is going on.*1831

*And then, we are going to come up with quick ways of doing the integration.*1836

*You have actually already done some of them, antiderivatives.*1840

*An antiderivative is the integral, that is what is going to be happening.*1843

*For now, we will evaluate the definite integral using the definition.*1848

*As such, we may need the following to help us.*1853

*We may need the following properties to help us.*1861

*If I have a sum i goes from 1 to n of i, there is a closed form expression for that.*1878

*It is going to be n × n + 1/ 2.*1886

*In other words, what this is saying is that, if I take, let us say n is 15.*1890

*If I say, out of the numbers from 1 to 15, 1 + 2 + 3 + 4 + 5 + 6 . . . + 13 + 14 + 15,*1894

*if I add all of those up, I can actually just take 15 × 16 and divide by 2.*1902

*This gives me a closed form expression for the sum, that is all this is.*1908

*Just some formulas that are going to help us along.*1914

*I have another one, the sum as i goes from 1 to n of i², that also gives us a closed form expression.*1921

*+ 1 to 1 + 1/ 6.*1930

*I happen to have a closed form expression for i = 1 to n of i³.*1936

*If I take 1³ + 2³ + 3³ + 4³ + 5³, however high I want to go, that is going to be n × n + 1/ 2².*1942

*Let us see b as a constant, if c is a constant and if I have the summation as i goes from 1 to n,*1963

*I have just plain old c, no i anywhere, that is just equal to n × c.*1977

*It is just telling me, i = 1, let us say n is 10, that means add c 10 times.*1985

*C + c + c + c and another 5×, that is 10 × c, n × c.*1991

*This is just a formula for, when I see this, I can substitute this.*1999

*That is all we are doing here.*2005

*It is just writing down some formulas that we can use, when we actually start evaluating these integrals using the definition.*2007

*Another thing that we can do is 1 to n c × ai, if I have some expression that involves a c, I can pull the constant out in front of it.*2015

*The same that I do, when I distribute.*2029

*When I factor things out, I can factor out this c.*2032

*Because this is just ca sub 1, ca sub 2 + ca sub 3.*2035

*They all involve c, just pull the c out.*2041

*It equal c × the sum i = 1 to n of a sub i. *2045

*What this means is that you can pull a constant out from under the summation symbol.*2052

*Let us do the sum as i go from 1 to n.*2089

*If we have a sub i + b sub i, let us say + or -, I can separate these out.*2095

*I can write this as, the sum i go from 1 to n of a sub i, + or -, depending on whether it is + or -.*2102

*In other words, the summation distributes over both.*2108

*You can think of it that way, b sub i 1 to n.*2113

*Properties of the definite integral.*2120

*Let us go to properties of the definite integral.*2123

*These are all going to be very familiar because integration is just a fancy form of summation.*2136

*The summation symbols, everything that we do for summation, we can do for the integral.*2142

*The integral from b to a of f(x) dx is equal to the negative of the integral from a to b of f(x) dx.*2154

*In other words, if I switch the order of integration, if I do here, my Δx is b – a/ n.*2166

*Here it is b – a/ n, here it is going to be a – b/ n.*2177

*If I integrate from b to a, instead of from a to b, all I do is switch the sign of the integral.*2184

*That is it, very simple, that is all.*2188

*In calculus, we are adding from left to right.*2193

*We have some function and we have broken up into a bunch of rectangles.*2199

*If I add in this direction from a to b, I get a number.*2202

*If I decide to add in this direction, I get the negative of that number.*2206

*That is all this is saying, switch the upper and lower limit, change the sign.*2210

*The integral from a to a of f(x) dx that is equal to 0.*2219

*The integral from a to b of cdx, c is any constant, that is just equal to c × b – a.*2227

*All this says is, if I have a constant function from a to b, the integral, it is just c, *2239

*the value of c which is this height × this distance, the area.*2251

*Or it could be negative because it is the integral, integral.*2257

*Now we are thinking, areas can now be negative.*2260

*If we introduce this notion of a negative area, it is not a problem.*2263

*Some further properties, the integral from a to b of c × f(x) dx, you can pull the constant out, c × the integral from a to b of f(x) dx.*2269

*The integral from a to b of f(x) + g(x) dx, the integral of a sum is equal to the sum of the individual integrals.*2288

*In other words, the integral sign distributes over both.*2301

*I will just write up the whole thing, of f(x) dx + the integral from a to b of g(x) dx.*2311

*The integral from a to c of f(x) dx = the integral from a to b of f(x) dx + the integral from b to c of f(x) dx.*2326

*This last one says I can actually break this up.*2341

*If I have a here and if I have c here, if i have b here, and if that is my function, *2345

*the integral from a to c is just the integral from a to b + the integral from b to c.*2353

*That is all I’m doing, just adding them up.*2358

*If f(x) is greater than or equal to 0 on the interval ab, then the integral of f(x) dx is greater than or equal to 0.*2370

*Very simple, nothing happening here.*2391

*Integration is something called an operator.*2396

*An operator is just a fancy word for do something to this function, in other words, operate on it.*2403

*If I give you a function f(x) and I say integrate it, integrate it means do something to it.*2408

*Take this function through a series of steps and spit out a number at the end.*2412

*When you see f(x) is greater than or equal to 0, you already know from years and years, elementary, junior high school math, *2418

*that whenever you have an quality or an inequality, as long as you do the same thing to both sides of the quality or the inequality,*2426

*you retain that equality or inequality.*2436

*Here, f(x) is greater than or equal to 0.*2439

*If I integrate the left side, integrate the right side, I get the left side of the integral of f(x) dx.*2441

*The integral that of 0 is just a bunch of 0 added together.*2449

*It is 0, it stays.*2453

*Think about it that way.*2456

*When you see an equality or inequality, you can integrate both sides.*2458

*It retains the equality or inequality, it retains the relationship.*2463

*Just like if you take the logarithm of both sides, if you exponentiate both sides, if you multiply both sides by 2,*2467

*if you divide both sides by 5, as long as you do it to both sides, you are fine.*2472

*Treat the integral as some operator, as some thing that you do.*2477

*If f(x) is greater than or equal to g(x) on ab, then exactly what you think.*2484

*The interval from a to b of f(x) is greater than or equal to the integral from a to b of g(x).*2499

*This is greater than that so the integral is greater than integral, very simple.*2508

*If m is less than or equal to f(x), less than or equal to M on ab,*2516

*then m × b - a less than or equal to the integral a to b, f(x) dx less than or equal to m × b – a.*2532

*If some function on this interval happens to have a lower bound and an upper bound, *2548

*then the integral of that function is going to be greater than the lower bound × the length of the interval.*2555

*It is going to be less than the upper bound × the length of the interval.*2563

*All we have done here is take the integral of this, the integral of that, the integral of that.*2567

*The integral of that, that is the symbol.*2571

*The integral of a constant m dx, we said it is equal to m × b – a.*2574

*The integral of M is M × b – a.*2579

*Just integrate, integrate, solve for the things you can solve and leave those that you cannot, as the symbol.*2587

*Thank you so much for joining us here at www.educator.com.*2596

*We will see you next time, bye.*2599

1 answer

Last reply by: Professor Hovasapian

Thu Dec 17, 2015 12:59 AM

Post by Gautham Padmakumar on December 12, 2015

42:00 did you forget to put the differential of x in the inequality