 Welcome back everyone. Let's take a look at an example of computing the volume of solid revolution using the disk method. And to describe the solid in question, take the following region you can see bounded between the x-axis and this parabola f of x equals 4 minus x squared. And let's rotate this around the x-axis and consider the following solid revolution and try to visualize this in your mind here. The region you can see here, and I would recommend as you're trying to work through these examples and exercises on your own, draw the picture of the two-dimensional region. It can be very difficult to draw or even visualize the three-dimensional region if you can appreciate the three-dimensional solid. If you can at least see the two-dimensional region that forms the solid, that can be very helpful in understanding exactly what it is you're trying to compute. So I mean this one when you get rotated, it's going to make somewhat of like a football-like shape with this parabola. Y equals 4x minus 4 minus x squared. This is going to be, you take your standard parabola, but it's going to be reflected downward and then you shift it up by 4, which you see right here. Now in this situation, you'll notice that if we're going to try to find the volume using the disk method, remember your volume, you would take the integral from A to B of pi r squared dx. The radius of the disk is r here and dx is the thickness of a single disk here. We have to also know the boundaries. A equals A equals, or x equals A, x equals B, what are the bounds right here? Now with a picture, we can see very clearly that we're looking for these values right here. We need to find the x-intercepts of this region here. And so we were looking for when f of x, which is 4 minus x squared, would this equal 0? There's a couple ways you could solve this. You could solve it by factoring. You could solve it for x. You could use the quadratic formula. Any of those are pretty good. I mean quadratic formula seems like overkill on this one. I mean 4 minus x squared, this looks like a difference of squares factorization to me. So we get x minus, or 2 minus x and 2 plus x are as our factors. And so our x-intercepts will be plus or minus 2. And so we're going to integrate this from negative 2 to 2 pi r squared. Well, we'll plug in the r in just a second. I want to pause for a moment and do make a comment here that notice your bounds. Your upper bound is a 2 and your lower bound is a negative 2. This is a symmetric interval. It's symmetric, meaning that the distance you go to the right of the y-axis is the same as you go from the left. And when it comes to integrals involving symmetric bounds, there's two things you should remember about this. So remember, if you go integrate from negative a to a and you integrate an odd function, that's always going to equal 0. Absolutely, because the area above the x-axis is equal in opposite to the area below the x-axis, so the area would be 0. Now for a problem like this, the volume of this football-like shape is not 0. So we're not going to be getting an odd function. On the other hand, if we take the integral from negative a to a of an even function, remember even functions are those functions which are symmetric with respect to the x-axis. This should equal two times the integral from 0 to a of that same even function, right? And so given the symmetry of the region here, how it's symmetric with respect to the y-axis, the solid of revolution will likewise be symmetric. And so we actually do anticipate to have this even symmetry appearing inside of this situation right here. And so let's proceed back to the integral I raised some of it. Whoops, negative two to two pi times the radius. How big is the radius, right? The radius of the disk, you can draw these cross-sectional rectangles right here to see what does a typical cross-section look like. And so the height of this rectangle right here is going to be the radius of the disk you're looking for. And notice that we're just looking for the distance between the x-axis and the function. That's just going to be the y-coordinate. So if you have this point right here, x comma y, we're just looking for that y-coordinate, which the y-coordinate is given as y equals 4 minus x squared. And so then we get 4 minus x squared as our radius. We square it and we get a dx right there. All right. And so as I predicted, this is in fact an even function. If you were to place x with negative x, this thing doesn't change whatsoever. And so for the sake of simplicity, what I'm going to do is actually double half of the volume. So I'm going to calculate the volume from 0 to 2 of the same function pi times 4 minus x squared dx. And so the reason I'm doing this mostly comes with the fact that later on I have to plug in the number 0 or negative 2. 0 is always much easier arithmetic. So I'm going to prefer using symmetry just to make the arithmetic simpler. It doesn't help the calculus at all, but the arithmetic is enough of a challenge, right? So let's go that way. So with this one, 4 minus x squared is quantity squared. There's no use substitution that's going to help us out here. So we really have no other choice but then just to foil out the 4 minus x squared squared. And so by the usual foil method, if we do that, we end up with 16, which is 4 squared. We're going to get negative 8x squared plus x to the fourth dx, like so. And then the typical power rule comes into play right now. I'm actually going to take the pi out as well. So we get 2 pi right here. And so we're going to get 16x minus 8 thirds x cubed plus x to the fifth over 5 as we go from 0 to 2. Now notice that when you plug in the 0, all these multiples of x's are just going to go to 0. So arithmetic with 0, which is really, really the nice part there. When we plug in the 2 on the other hand, there is some effort that's going to happen there. You're going to get 16 times 2, which is 32. You're going to get 8 thirds times 2 cubed, which is itself 8. And then we're going to get 2 to the fifth, which is 32. That sits above 5. And we can times the 8's together right here and end up with 64 over 3. But then we just have our left with adding some fractions together. And as that's just tedious, but trivial arithmetic, I'm going to skip over the details. And you can check this yourself. When you add these together, you're going to end up with 512 pi over 15 as the volume of this solid here. So in this example, some of the things to look out for is that in order to determine the region that is being revolved, you might have to find these implicit boundary points. We've seen things like that before when we talk about area between the curves. Also, you might want to use symmetry to help you simplify the integral calculations, particularly that arithmetic you see at the very end of the integral.