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Washer or ring method for vertical line rotation

Video transcript
What we're going to do in this video is take the region between the two curves, y is equal to square root of x on top and y is equal to x squared on the bottom and rotate it around a vertical line that is not the y-axis. So we're going to rotate it around the vertical line x is equal to 2. We're going to rotate it right around like that. And if we rotate it like that, we will get this shape, so this strange-looking shape. It's hollowed out in the middle. You can kind of use the y equals x squared part kind of hollows out the middle. And it's really the stuff in between that forms the wall of this rotated shape. So let's think about how we can figure out the volume. And we're going to do it using the disk method, sometimes called the ring method-- actually, it's going to be more of the washer method-- to do this one right over here. So we're rotating around a vertical line. We want to use our disk or ring or washer method. And so it'll be really helpful to have a bunch of rings stacked up in the y direction. So we're probably going to want to integrate with respect to y. Let me make it clear what I'm talking about. Each of our rings are going to look something like this. So let me do my best attempt to draw. So that's the inner radius of the ring defined by y is equal to x squared. And then the outside radius of the ring might look something like this. My best attempt to draw it reasonably. The outside radius of the ring might look something like that. And the depth of the ring would be a little dy just like that. So let me draw the depth. So let me see, right over here. You could kind of view it as the height of the ring, because we are thinking in a vertical direction. And that would be a ring for a given y for, say, this y right over here. It's essentially the ring generated if you were to take this rectangle of height dy and rotate it around the line x equals 2. And we're going to construct a ring like that for each of the y's in our interval. So you could imagine stacking up a whole set of rings just like this and then taking the sum of the volumes of all of those rings and the limit of that side as you have an infinite number of rings with-- or close to infinite-- or really, infinite number of rings with infinitesmally small height or depth or dy. So let's figure out how we would do this. Well, we know that all we have to do is figure out what the volume of one of these rings are for given y as a function of y and then integrate along all of them, sum them all up. So let's figure out what the volume of one of these rings are. And to do it, we're going to express these functions as functions of y. So our purple function, y is equal to square root of x, if we square both sides, we would get y squared is equal to x. And let me swap the sides so it makes it clear that now x is a function of y. x is equal to y squared. That's our top function the way we've drawn a kind of outer shell for our figure. And then y is equal to x squared. If you take the principal root of both sides of that and it all works out because we're operating in the first quadrant here. That's the part of it that we care about. So you're going to get x is equal to the square root of y. That is our yellow function right over there. Now, how do we figure out the area of the surface of one of these rings or one of these washers? Well, the area-- let me do this in orange because I drew that ring in orange. So the area of the surface right over here in orange as a function of y is going to be equal to the area of the circle if I just consider the outer radius, and then I subtract out the area of the circle constructed by the inner radius, just kind of subtract it out. So the outer-circle radius, so it's going to be pi times outer radius squared minus pi-- let me write it this way-- outer-- well, I'll just write it-- outer radius-- this is going to be as a function of y-- outer radius squared minus pi times inner radius squared. And we want this all to be a function of y. So the outer radius as a function of y is going to be what? Well, it might be easier to visualize-- actually, I'll try it both places. So it's this entire distance right over here, essentially, the distance between the vertical line, the horizontal distance between our vertical line, and our outer function. So if you think about it in terms of x, It's going to be 2 minus whatever the x value is right over here. So the x value right over here is going to be y squared. Remember, we want this is a function of y. So our outer radius, this whole distance, is going to be 2 minus the x value here as a function of y. That x value is y squared. So the outer radius is 2 minus x-- sorry. 2 minus y squared. We want it as a function of y. And then our inner radius is going to be equal to what? Well, that's going to be the difference, the horizontal distance, between this vertical line and our inner function, our inner boundary. So it's going to be the horizontal distance between two and whatever x value this is. But this x value as a function of y is just square root of y. So it's going to be 2 minus square root of y. And so now, we can come up with an expression for area. It's going to be-- and I'll just factor out-- actually, I'll leave the pi there. So it's going to be pi-- right over here-- it's going to be pi times outer radius squared. Well, the outer radius is 2 minus y squared-- and let me just-- well, I'll just write it. 2 minus y squared-- and we're going to square that-- squared, minus pi times the inner radius squared. Well, we already figured that out. The inner radius is 2 minus square root of y. And we're going to square that one, too. So this gives us the area of one of our rings as a function of y, the top of the ring, where I shaded in orange. And now, if we want the volume of one of those rings, we have to multiply it by its depth or its height the way we've drawn it right over here. And its height-- we've done this multiple times already right over here-- is an infinitesimal change in y. So we're going to multiply all that business times dy. This is the volume of one of our rings. And then we want to sum up all of the rings over our interval. So we're going to sum up all of the rings over the interval. And when you take the integral sign, it's a sum where you're taking the limit as you have an infinite number of rings that become infinitesimally small in height or depth, depending on how you view it. And what's our interval? So we've looked at this multiple times. These two graphs-- you could do it by inspection. You could try to solve it in some way, but it's pretty obvious that they intersect at-- remember we care about our y interval. They intersect at y is equal to 0 and y is equal to 1. And there you have it. We've set up our integral for the volume of this shape right over here. I'll leave you there, and in this next video, we will just evaluate this integral.