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

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

## Discussion

## Download Lecture Slides

## Table of Contents

## Transcription

### Areas Between Curves

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
- Areas Between Two Curves: Function of x
- Graph 1: Area Between f(x) & g(x)
- Graph 2: Area Between f(x) & g(x)
- Is It Possible to Write as a Single Integral?
- Area Between the Curves on [a,b]
- Absolute Value
- Formula for Areas Between Two Curves: Top Function - Bottom Function
- Areas Between Curves: Function of y

- Intro 0:00
- Areas Between Two Curves: Function of x 0:08
- Graph 1: Area Between f(x) & g(x)
- Graph 2: Area Between f(x) & g(x)
- Is It Possible to Write as a Single Integral?
- Area Between the Curves on [a,b]
- Absolute Value
- Formula for Areas Between Two Curves: Top Function - Bottom Function
- Areas Between Curves: Function of y 17:49
- What if We are Given Functions of y?
- Formula for Areas Between Two Curves: Right Function - Left Function
- Finding a & b

### AP Calculus AB Online Prep Course

### Transcription: Areas Between Curves

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

*Today, we are going to be discussing the areas between curves.*0004

*Let us jump right on in.*0007

*Let us look at the following, let me go ahead and work in blue today.*0011

*Let us look at the following situation.*0021

*We have ourselves a little bit of graph here.*0028

*We have one graph, let us go ahead and call this one f(x).*0034

*And then, we have another one let us say something like that.*0039

*We will just go ahead and call this one g(x).*0044

*Let us pick an interval from a to b.*0047

*We have got this and that.*0051

*The question is how can I find the area between f and g?*0057

*How can I find this area right here?*0075

*It is exactly what you think.*0079

*The area under f(x) is just this, it is just the integral.*0086

*As you know, from a to b of f(x) d(x).*0093

*Let me go ahead and do that in red.*0098

*That takes care of everything underneath f(x).*0099

*Let me go back to blue.*0107

*The area under g(x), that is just the integral from a to b of g(x) dx.*0108

*I will go ahead and do this one in black, that is this region right here, underneath g(x).*0119

*You know already for years and years and years now, this area is going to be the area of f - the area of g.*0126

*And that will give me the area that I'm interested in, this portion right here.*0133

*That is it, it is nice and simple.*0140

*The area above - the function of the area below, that is all.*0142

*Let me go back blue.*0148

*Since f(x) is greater than or equal to g(x) on all of the interval ab,*0155

*the area between is the integral from a to b of f(x) d(x) - the integral from a to b of g(x) dx.*0168

*Nice and straightforward.*0190

*That is going to be equal to, I will go ahead and put those integrals together.*0195

*It is going to be the integral from a to b of f - g dx, that is it.*0199

*Generally, you are going to express it like this, f - the g.*0207

*And then, as a practical matter, when you solve the integral, you are going to separate it out as integral of f - the integral of g.*0210

*This is when over the entire interval f is above g.*0219

*Upper function - the lower function, that is essentially how it goes.*0224

*Area equals the integral from a to b, we will write upper, the upper function - the lower function d(x)*0230

*or whatever variable you happen to be integrating with respect to.*0243

*What if we have the following situation?*0248

*Where we are over a given interval that actually cross.*0257

*What if we have the following situation?*0263

*We have got, we will call this one f(x) and then we can do something like this.*0273

*We will call this one g(x).*0281

*Let us go ahead and call this a, we have got that one.*0286

*We will just put b over here, something like that.*0294

*Now where they meet, I’m going to go ahead and call this c.*0298

*The area between the curves is going to be this area right here, this area and that area.*0306

*However, from a to c, it is f that is on top and it is g that is lower.*0319

*But from c all the way to b, now I have g is on top and f is the one that is lower,*0328

*where we find the area by taking the upper function - the lower function.*0336

*You have to split this up into two integrals.*0340

*You have to integrate from a to c doing f – g.*0342

*You have to do the integral from c to b of g – f.*0347

*That is all, it is that simple.*0353

*You just add the integrals together, you just have to separate them.*0354

*You have to find out where they meet, to find that x value.*0357

*How do you do that, you set the two functions equal to each other and you solve for x.*0361

*Let us go ahead and write this out.*0368

*On the interval ac, f is greater than or equal to g.*0371

*The area is actually equal to the integral from a to c of f - g dx.*0379

*Of course from c to b, here to here, in this particular case, it is g that is greater than or equal to f.*0389

*The area = the area from c to b of g – f dx, that is it upper – lower.*0400

*That is all you really have to know.*0413

*Whichever graph is higher up on a given interval is the first entry.*0423

*The first entry is the upper graph, entry in the integrand.*0456

*The integrand is the thing that is underneath the integral sign.*0468

*Because we want the integrand to be positive, because we are dealing with areas.*0474

*That is it, nice and simple.*0490

*A little bit of foray into some notation.*0498

*Is it possible to write the integral from a to c of – g dx + the integral from c to b of g - f dx.*0502

*The integral we just took, we split it up.*0526

*Is it possible to write that as a single symbol, as a single integral?*0528

*The answer is yes, the symbol for it is the following.*0536

*For f(x) and g(x) both continuous on ab, both continuous on the interval ab,*0547

*the area between the curves on of the interval is, the symbolism we use is the absolute value symbolism.*0565

*a to b absolute value of f - g dx.*0583

*The truth is you can actually do it in either way.*0589

*You can do f – g, g – f.*0591

*The absolute value sign, this is this.*0592

*I will show you why in just a minute.*0598

*This is the actual statement of how we find the area between two curves.*0601

*If you are given the curve f, given the curve g, you take the integral from a to b,*0607

*whatever interval you are dealing with of the absolute value of f(g).*0612

*The absolute value of f(g) is actually telling you to do something.*0616

*When we solve these, we do not use this, the symbolism asks us to do something.*0621

*What the symbol is telling us is to actually separate it out, here is how.*0626

*Let us revisit absolute value.*0631

*I find that kids, that absolute value is one of the things that kids know how to do*0636

*but they do cannot really wrap their minds around what an absolute value is saying.*0645

*Let us revisit it, it is always good to revisit it a couple of times.*0651

*For some odd reason, absolute value always is a little, people are not quite sure how exactly to go about it.*0656

*We will discuss it now, let us revisit absolute value.*0663

*The definition of absolute value is the following.*0668

*The absolute value of a, whatever is between that absolute value symbols is equal to the following.*0674

*It is equal to just a, if a, the thing in between the absolute value size is bigger than 0.*0683

*But it is equal to –a, if a, what is in between the absolute value sign is less than 0.*0694

*If whatever is in between the absolute value signs is bigger than 0,*0702

*it just means drop the absolute value signs, take the number as is.*0706

*In other words, what is the absolute value of 5, it is just 5.*0709

*If what is in between the absolute value signs is negative, then the value of the absolute value sign is negative of a.*0714

*In other words, if I had the absolute value of -5, the definition says take - -5, that is my answer which I know is 5.*0726

*You do it automatically.*0737

*But when you see it in the context of something like this in integral, for numbers it is fine.*0739

*You know that the absolute value of -5 is 5.*0743

*What is the absolute value of f – g?*0747

*Let us look at f – g.*0751

*Once again, this is what is important, this definition right here.*0760

*If the thing between the absolute value sign, the whole thing,*0766

*whether it is a number, a letter, or an expression, if it is bigger than 0, then we just take expression as is.*0768

*If it is less than 0 then we take the negative of the expression as is.*0776

*Now we have f – g, just like our definition.*0789

*I will actually do this in reverse.*0798

*If f – g, f – g, what is the absolute value of f – g?*0801

*If what is in between the absolute value signs, if f - g is less than 0 which is equivalent to saying f is less than g,*0810

*then the absolute value of f - g is just plain old f – g.*0822

*If f - g is less than 0 which is equivalent to saying, sorry I have this backwards,*0834

*if f - g is greater than 0, the thing underneath the absolute value signs*0848

*which is equivalent to saying f is greater than g, I just move this g over here.*0852

*If f is greater than g and the absolute of f - g is just f – g.*0857

*If f - g is less than 0 which is equivalent to saying that f is less than g or g is bigger than f,*0862

*then the absolute value of f - g is - f – g.*0872

*What is f - f – g, it is g – f, that is it.*0879

*That is all the absolute value symbol is saying.*0882

*In the case of an expression, you are going to negate.*0884

*If that expression is less than 0 then you negate the entire expression.*0888

*When you negate a difference, the term is flipped and you are getting that.*0893

*This is just a symbolic way of representing what it is it that we did, in terms of two separate integrals.*0897

*The absolute value symbol accounts for all cases.*0910

*It is just a shorthand notation, all cases all on the interval ab.*0924

*When we actually do the integration, in practice, we still just separate*0938

*the area of calculation into two or more areas, depending on how many times it crosses.*0954

*In each case, we always take the upper – lower, upper – lower, upper – lower.*0971

*The symbol, the integral from a to b, the absolute of f – g dx, it just gives us a compact notation.*0984

*It actually tells us that if f - g ever drops below 0, I have to switch those.*0999

*That is what the absolute value symbol is telling me.*1007

*It just gives us a compact notation, there we go.*1010

*The area of a region between two curves is the integral from a to b of the upper function - the lower function.*1027

*It is probably the best way to think about it, dx, that is all.*1054

*What if we are given functions not in terms of x but in terms of y?*1061

*Now instead of x being the independent variable, what if we are given something like this?*1066

*What if we are given functions of y?*1074

*For example, x = y², let us say the other function is x = y – 2/ 2.*1084

*They are going to ask, what is the area between these curves?*1102

*We are accustomed to seeing y in terms of x.*1107

*Here we have x, in terms of y.*1110

*Now y is the independent variable.*1113

*Whatever y happens to be, we do something to it and we spit out an x.*1115

*Let us graph these two and see what we are dealing with.*1120

*x = y², whenever you flip x and y, the role of x and y, what you have done is actually take the inverse function.*1131

*If I know that my normal x, x², y = x² is my parabola that looks like that.*1138

*My x = y² is my parabola that looks like this.*1145

*It is just moving along the x axis, instead of the y axis.*1152

*Let me erase these little arrows, it is confusing.*1156

*x = y – 2/ 2, let us go ahead and put in the form that we are actually used to seeing it, as far as lines are concerned.*1160

*I'm going to multiply by 2 and you are going to end up getting y is equal to 2x.*1167

*I’m sorry, this should be +2 – 2.*1180

*I multiply by 2, I’m going to move that 2 over, and I get y = 2x – 2.*1184

*It is okay, we can do that.*1189

*We can flip it around, in order to help us graph it.*1190

*I can do the same thing here, if I wanted to.*1193

*This is going to be y is equal to + or -√x which I know is this curve and is this curve.*1196

*That is fine, you can go ahead and do that, if I need to graph it.*1204

*Now y = 2x – 2, let me come down and mark -2.*1208

*It is up 2/ 1, up 2/ 1, I’m going to get basically a line that looks like that.*1213

*The area that I'm interested in is this area.*1221

*Notice what we have here, we have an upper function, we have a lower function.*1229

*There is a bit of an issue here.*1240

*It is like from 0 to whatever this point happens to be, this is the upper function and this is the lower function.*1241

*But from here to this x value, this is my upper function, my line is my lower function.*1250

*If I were to integrate this along x, in other words, make a little rectangle like that and add this way,*1260

*from your perspective, this way, moving in this direction, I have to break this up into two integrals.*1268

*In this case, it is actually better to integrate along y.*1277

*In other words, along the axis of the independent variable.*1282

*Here the independent variable is y.*1286

*It is best to integrate along y.*1288

*When you are given a function of x, it is best to integrate along x.*1291

*What happens here is the following.*1296

*Whenever you are given functions in terms of y, the formula becomes,*1302

*the area is equal to the integral from a to b.*1321

*This time it is going to be the right function - the left function.*1325

*Before we have upper – lower, now we have right – left.*1333

*And of course, we are going to be integrating along the y axis, so it is dy.*1337

*But what are a and b?*1343

*If we are integrating along y, they are the points on the y axis.*1346

*Let us go ahead and write it out.*1354

*But what are a and b?*1356

*We want the area between the curves.*1368

*a and b are just the y values of the points where the two graphs meet.*1387

*Points are just the y values of the points where f(y) = g(y).*1403

*In other words, you do what the same thing that you do any other time.*1412

*You set the two graphs equal to each other, you see where they meet.*1415

*But now instead of taking the x values, you take the y values because we are integrating with respect to y.*1419

*Let us go ahead and do this problem.*1425

*Let us do this problem.*1435

*We had this graph, let me check something real quickly here.*1437

*y = 2x – 2, that is fine.*1457

*We hade this graph where we have this and we have this line.*1469

*This was our x = y² and this one we had y = x – 2.*1476

*When you set them equal to each other, you can do it, you got x = y².*1487

*Let me do this in red.*1497

*x = y², and then we have this other version of it, in order to make it easier for us to actually graph.*1502

*y = x – 2, you can go ahead and put the x - 2 in here.*1509

*I should do it this way.*1519

*I have x = y² and this becomes y + 2.*1522

*I think I’m getting ahead of myself, let me go back to blue.*1530

*Let me rewrite down my functions properly.*1538

*This is y = 2x – 2.*1541

*This function = x = y².*1542

*We are looking for the area that is contained here.*1545

*What I'm going to do is I'm going to find that point and that point.*1551

*I’m going to find the y values of that point which are here and here.*1556

*That is going to be my a and that is going to be my b.*1560

*That is what is going on here because we are going to be integrating along the y axis now,*1563

*taking the right function which is this one.*1567

*This is the left function.*1572

*Let us go ahead and see what we were dealing with.*1577

*We have got x = y².*1580

*I have y = 2x – 2, I got y + 2 = 2x.*1585

*I have got x = y + 2/ 2.*1593

*x = y², x = y + 2/ 2.*1598

*I got y² = y + 2/ 2.*1603

*When I solve this, I’m going to get two values of y.*1608

*Move this over, turn it into a quadratic.*1612

*I’m going to get two values for y.*1614

*The y values that I get, those are my a and b.*1618

*That is exactly what is happening here.*1622

*Let us take a look at this, I went ahead and I use mathematical software to go ahead and graph this for me.*1624

*You can use your calculator, any kind of online software that you want.*1633

*In this particular case, I use something called www.desmos.com.*1639

*It is available the minute you pull it up, you click this big red button that says launch the calculator.*1645

*This screen comes up and you can actually do your graphs.*1649

*That is what I use for all of the pictures that I generate here.*1654

*When I do this, x = y² and I just wrote it as y = 2x – 2.*1657

*When I graph this, I end up finding this point and this point.*1661

*Let me go ahead and go back to blue.*1669

*My y values are 1.281 and -0.781, that is here and here.*1670

*The area equals -0.781, the integral from -0.781 negative to 1.281 of the right function*1682

*- the left function, expressed in terms of y.*1698

*That was going to be the right function, this one, in terms of y.*1702

*Here we have y = 2x – 2.*1711

*It is going to be y + 2/ 2 is equal to x.*1713

*It is going to be y + 2/ 2 - the left function.*1719

*2 - y² dy, that is it.*1738

*Because I'm integrating vertically like this, taking a little horizontal strips that way,*1748

*it is going to be integral from this point to this point, that is my 0.781 negative to 1.281 positive of the right function,*1758

*expressed in terms of y which is y + 2/ 2 - the left function, expressed in terms of y y² dy.*1768

*In the problems that we are going to do which is going to be the next lesson,*1780

*when you are given a set of functions, you are just going to be given functions randomly.*1783

*Sometimes they are going to be in terms of x, sometimes they are going to be in terms of y, you do not know.*1797

*When you are given a set of functions and ask to find areas between regions,*1803

*you will have to decide what is going to be the best integration.*1826

*I’m going to integrate this along the x axis and I’m going to integrate along the y axis.*1830

*What is going to be the easiest?*1836

*Sometimes you can do both, but one of them is longer than the other.*1837

*Sometimes it is best only to do one, either along y or along x, you get to decide.*1841

*When is it not necessarily the form of the function, the only thing we have done is say that,*1847

*if you are going to be integrating along x, you are going to be taking the upper function - the lower function and integrate it.*1852

*If you are going to be integrating along y, you are going to be taking the right function - the left function.*1859

*You, yourself, have to decide which one is best and decide how to manipulate the situation, according to what is best.*1865

*It is not necessarily some algorithm or recipe that you want to follow.*1872

*You want to take a look at the situation and decide what is best.*1876

*In this particular case, it was best to just go this way because you have a left and a right function, and a left function.*1879

*You have a series of rectangles that touch both functions.*1886

*If you were to decide to do this with respect to x, which you can, you have to break it up here.*1890

*You have to integrate from here to here, this being your upper function, this being your lower.*1897

*And then, you have to integrate from here to here.*1903

*This being your upper function, this being your lower function.*1906

*We are going to do that in just a second.*1910

*You will decide what is best.*1912

*You will decide the best way to integrate.*1919

*Let us go ahead and actually do it the other way.*1935

*What will this integral look like, if we decided to integrate along the x axis?*1937

*First, now we are going to integrate along the x axis.*1943

*It is going to be dx, if it is going to be dx, we need the functions to be expressed in terms of x.*1952

*We have got y = √x, y = -√x.*1958

*We already have this one, in terms of y.*1966

*y = 2x – 2.*1968

*However, we have to break it up.*1971

*Our first integral, from here to here, we are going to have rectangles.*1976

*This is going to be one representative rectangle for that area.*1982

*This is going to be a representative rectangle for that area.*1986

*We are going to need the top function - the bottom function.*1989

*The area is going to be the integral from 0 to 0.61 because that is the x value of where they meet.*1993

*From here to here, upper – lower.*2003

*It is going to be √x - - √x dx.*2007

*I’m going to add the second area.*2016

*It is going to be 0.61 to 1.64, upper function is √x - the lower function which is 2x – 2.*2019

*This one is only slightly longer; not more complicated, it is just slightly longer.*2040

*Again, you can do it both ways.*2046

*Ultimately, it comes down to a personal choice.*2049

*You have noticed with calculus that as the problems become more complicated, there are more ways of approaching it.*2051

*You get to decide what is the best integration.*2057

*Do not feel like you have to do one or the other, it is whatever you feel comfortable with, whatever your eye sees.*2060

*If you prefer to stick with dx and it is not too complicated, great, go ahead and stick with dx.*2066

*But sometimes you are not going to be able to do it with respect to x.*2073

*We will get into more of those problems later on.*2075

*Sometimes, you have no choice but to do it along the y axis, because the x integration is just going to be too complicated.*2077

*That is all, thank you so much for joining us here at www.educator.com.*2085

*The next lesson is going to be example problems for areas between curves.*2090

*Take care, see you next time, bye.*2094

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