 So far we've calculated volumes of solids of revolution by examining these cross-sectional discs, aka the disc method. It often happens that our discs are going to have holes in them. Instead, so if we think of like a hardware example, we think of these as kind of washers instead. So what I mean is something like the following situation. We have the disc that we had from before, but now there is this hole in it. And so this so-called washer is actually a disc with a concentric disc removed from it. And this is actually a fairly simple problem to adapt to. I'm going to draw my washer a little bit bigger here. So we're gonna have a really, again, draw really wide just so we can see a little bit. And the thickness of this washer disc, you should think it's really small. I'm just enlarging it for the sake of illustration right here. And so then we have this concentric disc that's been removed from the inside. And so this kind of continues down here, something like this. So we have this washer, again, just something like you would get from Home Depot or Lowe's or whatever, wherever you like to buy your tools and things like that, right? So we have this this washer, which is a disc with a disc removed from it. And so if we look at the common center of these two discs, there is going to be this outer radius, this outer radius that we see right here. And then there's going to be this inner radius associated to the smaller one, the inner radius. And the volume of a washer is going to be fairly simple, right? You're going to take the big washer, so the outer washer, you're going to take its volume. And you're going to subtract from it the inner, not I shouldn't say the inner, the outer washer should be the outer disc. And then we take the inner disc and remove it. And although removing your inner child from your life might be detrimental for your mental health, this is a perfectly good thing to do for washers and things like that. You take away the inner disc from the outer disc. Well, the outer disc has a volume of a pie we're going to take outer radius squared. It's thickness we'll call it delta x. And then we subtract from that pie times the inner radius I r squared times the thickness again we'll call it delta x. And so notice that this expression that both terms are divisible by pie we can factor that out. Both of them have the exact same thickness this delta x, which we're seeing right here, delta x. And so if we factor this thing, we end up with the following. We get pie. Well, actually, it's just written right here. We end up with pie times the outer radius squared minus the industry squared times f of x. Now suppose that the outer radius is given by some function, some function value, we'll call it f of x. And then the inner radius is given by some other function, which we call g of x. Then what this translates to happening is that for a single washer, we're going to have pie times f of x i star squared minus g of x i star squared times the thickness delta x. So this right here represents the volume of a single washer. We add these together to give an approximation of the volume of the entire solid of revolution. And to improve the approximation, we take more and more and more slices, more and more cross sections, taking the limit as n goes to infinity. This will give us the true volume of the solid of revolution. And this gives us the following formula. The volume would equal the integral from a to b pie times f of x squared minus g of x squared dx. Notice this is a difference of squares. We're not taking f minus g quantity squared. We square the f, we square the g and then we subtract it. The order of operations does matter there significantly. And so this is what's commonly referred to as the washer method. This is a generalization of the disc method we had seen before. It's just this now accommodates for the fact of what if our disc has a hole in it? We can think of the disc method as a special specification, a specialization of this washer method where what if the inner radius is zero, you remove nothing from it. So the two are often going hand in hand. And so let's illustrate this with an example here. So let's find the volume of the solid of revolution obtained by rotating this region that you see in front of you. It's bounded by y squared equals x and x equals two y about the y axis. So let's spin this thing around the y axis as you can see right here. And the region is given to you. And we can see this right here. And it's again, it's always to my recommendation that you draw these pictures out, try to visualize the solid the best you can. So as we rotate this thing around the y axis, we're going to get some type of conical shape as we rotate the x equals y or the x equals two y. But then there's going to be this curvature that happens in the inside of it. Something like that is the horrible drawing here. Let's try this one more time. We're gonna get something like this again. It looks like an ice cream cone. But then there's this. It's got this curve going on in the middle. Again, hideous drawing, you can see why I borrow three dimensional drawings from the textbook on these ones here. But that's the type of the solid we're trying to create right here. Now look at this cross sectional rectangle right here. You'll notice that we're trying to go from our axis, I'm gonna switch my color here, our axis is the y axis. And so we're trying to measure the distance coming out from here. So this distance you see illustrated right here, this is what we mean by our outer radius. The distance from the axis of revolution to the outermost point on our rotation. On the other hand, the inner radius is going to be much smaller distance. It's the distance from the y axis up to this point right here on the inside. The inner radius like so. And so this thing is going to have a hole. There's gonna be a hole inside of this solid or revolution. Because as we rotate this one, we rotate this one rectangle throughout, we're gonna make this washer like object that you see now illustrated here on the screen. So the washer method is going to be the appropriate technique to calculate the volume of the solid or revolution. So the volume is going to equal the integral. We'll come back to the boundary in just a second. If we apply the definition of the washer method, we're going to get pi times the difference of squares, we take the outer radius first. And so the one that's farther away from the x axis is going to be the line x equals two y. And so we're going to get two y squared minus the inner radius, which in this example would be the parabola x equals y squared. So we're going to go y squared squared. And then we have the thickness how thick are our rectangles? How thick are our cross sections going to be? As you see here in the illustration, the thickness of this rectangle because it's actually set horizontally, the thickness of the rectangle is going to be the y coordinate. It's a there's a small change of the y coordinate. There's a rise going on there. And so what that means for our integrals that our integral is going to have the differential dy. We're going to integrate this thing with respect to y. And because we're in respect to y, we want these to be functions of y, not functions of x. So you'll notice that in this situation, how we have x equals two y and x equals y squared, that is extremely preferable given that we want to integrate with respect to y. With the previous examples we've seen where our thickness was a dx. That tells us that we want to integrate with respect to x. And so we do have to pay attention to this differential. Because if we want to integrate with respect to x, instead of respect to y, this would have to become y equals the square root of x. And this one would have to become y equals one half x. So the approach you take depends on which variable you're going to integrate with respect to. So you're going to see the future we're going to pay attention to how thick are our cross sections. This emphasizes why it's so important to consider what does a typical cross section look like here. Alright, this gets us back now to the bounds of the integral. These are going to be y coordinates y equals whatever to y equals whatever. So for the y coordinate going up is the positive direction. So we need to figure out what is the y coordinate of this point right here. By illustration, this is clearly the origin 00. Notice if y equals zero x, you know, two times zero and zero squared is both zero. So that's the point of intersection. This point right here might be a little less intuitive. So let's solve it just by intersecting that is set the two equal to each other. We solve the equation to y equals y squared. You can subtract two y from both sides y squared minus two y equals zero. If you factor you get y times y minus two equals zero. And so the two points of intersection is zero, which we already knew and two is the other one. So this point right here is four comma two. Notice that two times two is four and two squared is likewise four. So we're going to integrate from y equals zero to y equals two. Notice that these are going to be the y coordinates, not the x coordinates, we want to see how the y changes because we have to determine where does the where do these rectangles live they live between y equals zero and y equals two. So once you set up the integral and for these type of story problems, that's always the hardest part setting up the integral correctly. Once you set up the integral correctly, the calculation is going to be fairly routine compared to what we've seen before. Just apply the fundamental theorem calculus at the right spot. So we have to algebraically prepare this thing to y squared will become a four y squared, and then y squared squared gives us a y to the fourth. This thing is now perfect for the power rule to go about. So we're going to get pi times four thirds y cubed minus y to the fifth over five, as we go from zero to two. We love the fact that the bottom numbers is zero because when we plug in zero, everything will just vanish. When we plug in two, we're going to get something non trivial. So let's see what that is. We get pi times four thirds times two cube, which is an eight minus two to the 30, two to the fifth, which is 32, all over five. Eight times five, of course, that's likewise a 32. You can factor out the common numerator of 32. So we end up with 32 pi. And this it's above now a one third minus a one fifth. Now to find a common denominator, we're going to have to times the one third by five over five. We'll do that for the one fifth as well, three over three, like so. So we're going to get five minus three for the numerator, which gives us another two over 15. And so then the final result should then be 64 pi over 15 as the the volume of this solid of revolution. And this gives us an illustration how one can use the washer method, which is a very nice generalization of the disk method.