AP Calculus AB 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 do0636

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 signs0848

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 separate0938

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 function1682

- 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