 This lesson is on volumes using discs and washes or volumes of revolution. Now don't forget anything that you have learned in the last lesson because some of that will be useful when we work through this lesson. So let's go on. Let's talk about discs revolving around the x-axis. So let's get a function f of x and revolve it around the x-axis. And once we do that, we have something that looks like this. I would say practice practice practice these until you are comfortable drawing these functions and making them look three-dimensional. In the last lesson, you sliced all your objects up. And we're going to do that again here. If we slice this up, we see that we have many many slices here and the area of each of those slices is nothing more than a circle. And the radius of our circle is f of x. So if we were finding the area, it's equal to pi r squared. And in our function notation, it would be pi f of x quantity squared. Now let's look at this in Riemann's some form. If we wanted the volume, we would say the volume is equal to pi times the limit as n approaches infinity of the sum from i equals 1 to n of f of x sub i quantity squared delta x sub i depending on which of these little discs that you wanted. Now, of course, your interval will be from some a to some b. And in this case, we'll make the a at the origin and b some other place on our curve. So once we do that, we can create the volume using our integral, which will be pi integral from a to b of f of x quantity squared dx. So this will be the formula that we will use in a plane disc problem when we revolve it around the x-axis. Here's an example. Find the volume when the region bounded by the curve y is equal to x squared. The x-axis from x equals 0 to x equals 4 is revolved around the x-axis. So what do we have? We'll get y is equal to x squared. Looks like that. We have the x-axis here from x equals 0 to x is equal to 4. So this is the region we are revolving about the x-axis. And when we do that, we'll have the mirror image here come down and we'll make it look three dimensional by putting the circle in there. So this is our function y is equal to x squared revolved around the x-axis. So what is our radius? Well, our radius is really y, which is equal to x squared. And if you think about the last lesson and we're looking for that area, so area, which is equal to the pi r squared, in this case will equal to pi times x squared squared, which is equal to pi x to the fourth. So if we put in volume formula, we will do volume is equal to the integral from 0 to 4 times pi x to the fourth and of course the little thickness, which is the delta x. If we do the math on this, we get pi times x to the fifth over 5 and we're going from 0 to 4. Once we compute that, we get pi times 4 to the fifth over 5 minus 0. So that is equal to 1024 pi over 5. Very simple problem, but always remember you always look for that radius even as they are getting more complicated. Let's go on to a different idea. Let's talk about revolving around the y-axis. So let's take our function and this time I'm going to say that our function is f of y. And we're going to revolve it around the y-axis so it looks more like this. And again, the region we are looking for is this region here and we'll call the initial point c in the ending point d. So in this case, our volume will be pi times the integral from c to d of f of y quantity squared dy. I didn't go through the Riemann sum on this because it's pretty redundant. And you'll see we're just changing from x's to y's on these. So our orientation is quite different. So let's do an example. Find the volume when the region bounded by the curve y is equal to x squared and the y-axis from y equals 0 to y is equal to 16 is revolved about the y-axis. So we have again y is equal to x squared from y equals 0 to y is equal to 16 in the y-axis. So that's this region in here. And it's very important on these that you know what region you're looking at and you revolve it about the y-axis. And remember if we're taking one slice out of this, it looks like that and this is our slice. And of course the radius in this case is this length here, which is x. So x in this case is equal to the square root of y. So our volume now is equal to pi times the integral by going from y is equal to 0 to y is equal to 16 times the square root of y quantity squared. Again, the thickness is dy. And of course the square root of y squared is y. So we have pi times the integral from 0 to 16 of y dy. And of course this is equal to pi times y squared over 2 from 0 to 16. And finalizing that we get 128 pi. And that is our final answer on that one. Let's go to another concept, washers. When we're looking at washers, if you think of a washer that you have that would ordinarily go in some sort of a faucet, it looks like that. So if we look at some sort of a picture of this, let's say we have two functions, one curve like that, maybe the other one aligned. And we want to revolve it about the x-axis. We do that mirror image down here. Again, practice, practice, practice. And we have an object that looks like that, a three-dimensional object. And when you actually begin to look at the circles, we have this inner circle. And then we have an outer circle. And you can see when I create the outer circle, I created a washer. So let's call the outer curve f of x and then that inner curve g of x. So how do we find the volume of what really is this region, this tiny region, worked around in a circle? Well, what we do is we find the volume of the outer function and minus the volume of the inner function. So in this case, this volume is equal to pi. The outer function is our f of x, so that's going to be f of x quantity squared. And then the inner function is g of x, so we'll have g of x quantity squared. Again, the little thickness for each of our circles. And let's say, again, we are going from some point a to some point b on this. So this is our general formula for finding the volume of a washer when it is revolved around the x-axis. Let's look at an example on this. Find the volume when the region bounded by the curves y is equal to x squared and y is equal to x is revolved about the x-axis. x squared looks like this. x will be some sort of line. And we have this point a and we have this point b, which is the intersecting point, which we all know to be one-one. And when we revolve it around the x-axis, it will look like this with the curve and then the little circular part there. If we just take a slice of this, again, we will have that washer where we have that inner curve and then the outer curve in that washer. So what's the volume? Again, the volume is pi times the integral of the area of the outer curve minus the area of the inner curve. So volume is equal to pi. In this case, the integral goes from zero to one. The outer curve is our x, so it's going to be x squared. Our inner curve is x squared. So if we square that, we get x to the fourth and that is all dx. And if we continue on, we get pi times x cubed over three minus x to the fifth over five from zero to one. Computing all that, we get two pi over 15. So again, look for what that radius is and when you are doing washers, it is the volume of the outer curve minus the volume of the inner curve. Now, suppose we did this around the y-axis. Again, the theory is pretty much the same. It's in y's this time, so let's just go straight to an example. Find the volume when the region bounded by the curves y is equal to x squared and y is equal to x is revolved about the y-axis. Same curve. This time, I'm just going to take the piece y is equal to x squared, y is equal to x, that piece there. Revolve it about the y-axis. Looks like this. Put in the circular piece. And again, we see we have a washer. We have the inner curve and we have that outer curve. Now, because our orientation is around the y-axis, our function, remember, has to be in y's. So, instead of saying y is equal to x squared, we have to say x is equal to the square root of y for that function. And of course, the line still is x is equal to y. Volume, again, is pi times the integral of the outer function. Now, the outer function in this case is the square root of y. So, it'll be the square root of y, quantity squared, minus the inner function, which is the line y squared. And all that is d-wide. An intersecting point, again, goes from y equals 0 to y is equal to 1. So, computing that, we get pi times y squared over 2 minus y cubed over 3. All from 0 to 1. And a final answer of pi over 6. Not too bad, just make sure the orientation is in the y direction and everything is in y's. Well, let's go to some other variations on this theme. Find the volume when the region bounded by y is equal to x squared, the y-axis, and y is equal to 16 is revolved about the x-axis. Well, this time, yes, we have our y is equal to x squared, that the y-axis, y is equal to 16 is up here. So, this is the region we want and it's revolved around the x-axis. So, it will look more like this when we finish it off. Now, normally, when you do these, your radius is just the curve itself. But this time, if you look, we have a washer. There's our inner curve, there's our outer curve. So, we have to do outer minus inner volumes. So, this time, our volume is equal to pi times the integral since it's going around the x-axis. Your interval is in x's, so this goes from x is equal to 0 and when y is equal to 16, x is 4. The outer curve is 16, so it's the distance from the axis to that line. So, it's 16 squared and some people think of this as a cylinder and you can just take the volume of a cylinder and subtract out the inside and that's fine too. Minus the inner function and that is y is equal to x squared, squared. So, that's what x to the fourth and that's all dx'd. And when we compute all of that, we will get 4, 0, 9, 6, pi over 5. We'll take the anti-derivative, etc., etc., so, and you would get that answer. But setting it up is the most important thing and even on the AP exam, you will see that sometimes they only ask you for the setups on problems like this. Let's go on to another variation. Find the volume when the region bounded by y is equal to x squared, the x-axis and x is equal to 4 is revolved about the y-axis. Let's find out what region we are looking at. y is equal to x squared, the x-axis, x is equal to 4, revolved about the y-axis. So, instead of finding the volume of the area near the y-axis, we're finding it away. And again, that leads us to that washer idea. There's the inner one, there's the outer one. So, this time we have to be in y's. So, again, we're going to change our x to be the square root of y and this length here for the outer one is 4. So, our volume is equal to pi integral 4 squared minus the square root of y squared. And of course, all that is d-wide and because we are in y's, we're going to go from y is equal to 0 and, of course, it ends at y is equal to 16. Computing all that, taking the antiderivative, et cetera, we should get 128 pi. So, again, setup is important. The integration part is quite easy on these, so there's no need for me to be running through those with you. But this is how you would set it up. Outer curves, volume minus inner curves, volume. Let's go to something else. Revolving about a line parallel to the x-axis, these are set up very interestingly. Find the volume when the region bounded by y is equal to x squared from y equals 0 to y is equal to 16 is revolved about the line y is equal to 16. Let's see what we have here. We have this y is equal to x squared from y is equal to 0 to y is equal to 16. So it's this region in here and we're going to revolve it about the line y is equal to 16. So it's all of that. So how do we do this? Well, what we actually do is move the curve down to the axis. So it looks like this. Simple technique when you are doing these. So what does it mean to move it down? Well, we're going to move it down 16 units. So our new radius is y minus 16. This y here, of course, minus 16, which is really x squared minus 16. We have just moved it down. So when we are doing our volume formula, we do pi times the integral of x squared minus 16 quantity squared pi r squared dx. And since we are doing both sides, we are going from negative 4 to 4 or we could go from 0 to 4 and multiply it by 2. If we finalize that answer, we have 2 times 8192 over 15 pi. Let's try another one. Revolving about a line parallel to the y-axis. Find the volume when the region bounded by y is equal to x squared. The x-axis and x is equal to 4 is revolved about the line x is equal to 4. Here we go. So here's y is equal to x squared, the x-axis, the line x is equal to 4, and we're going to revolve it around x is equal to 4. So it looks like this. Again, we are just going to move this. But because we are revolving around a line parallel to the y-axis, we have to change our y is equal to x squared to x is equal to the square root of y and move that. So volume is equal to pi moving square root of y minus 4. Remember we're moving left 4. If we were on the other side, we'd be moving right 4 and we're going to square that and our little dy there. And of course we have to go from 0 to 16 in order to get the top most point of this particular curve. And when we finalize that answer, we get 128 pi over 3. Remember, move it. This is the simplest way to think about these things. Some people think about the outer R versus the inner R and doing it from that point of view. But I think if you just think about moving these, it is so much easier. It's a transformation versus looking at it almost from the point of view of when we do a washer. So you don't ever get confused when you know that you have to move them. OK, well, here's one for you to try. Find the volume where the region bounded by y is equal to x squared. The x-axis and x is equal to 4 is revolved about the line y is equal to 16. And you will find your answers in your notes for this particular lesson. Hope this has gotten you through every different type of problem you can encounter when you are doing volumes using discs and washer. This is the end of your lesson.