 Hi everyone, we're going to work through an example of how to find the volume of a solid of revolution by using the washer method. Of course the washer method is similar to the disk method except for that hole in the middle. So in this problem we have a region bounded by a cubic function y equals x cubed. The y axis and the line y equals one and we are revolving that region about the line x equals one that vertical line and we are asked to find the volume of the resulting solid. So go ahead and graph it on your graphing calculators. I've already gone ahead and done that and taken a snapshot for you and I have adjusted my window to make it look nice for us. So if you go back to the problem and read about the region we're talking about, so we're bounded by the blue curve which is the cubic. The y axis over here on the left on the top side that horizontal line at y equals one and our axis of revolution is the vertical line x equals one so let's maybe draw that in if I can draw a vertical line at all. All right so this is x equals one and that's our axis of revolution. So the region that we're talking about is the one that you see in yellow. So if we want to take that region and revolve it around that vertical line x equals one imagine taking just that yellow part and flipping it over that vertical line so when we do that you know it kind of flops over to here sort of like that you know very very roughly. All right so you can try to imagine in your head what the resulting solid might look like. So with the washer method our representative rectangle similar to the disc method is perpendicular to the axis of revolution so in this case it's going to be going horizontally so this is going to become a dy problem but notice the representative rectangle is not touching the axis of revolution. All right that is the major difference between the washer method and the disc method. So if you think about the basic setup as we talked about in the lesson we know that our formula if you want to think of it that way is going to be pi times big r squared minus little r squared. Big r squared is going to be the distance from the axis of revolution to the far side of the rectangle. So that's going to be this part right here. All right so let's maybe come up with an expression for that first. All right and we're going to use this right minus left idea. All right so if we want an expression for a big r on the right it's hitting the vertical line x equals one on the right on the left it's hitting the y-axis. So we're doing right minus left on the right it's hitting the vertical line on the left it's hitting the y-axis so that capital R is one. Now for little r now little r is the distance from the axis of revolution to the close side of the representative rectangle so little r is going to be this little part in here it's basically the radius of the hole. All right so once again we're going to use this idea of right minus left so on the right little r is hitting the axis of revolution on the left though it's hitting the curve which is x cubed but it's a dy problem so we cannot use x cubed we need to put it in terms of y so we can't use that so we have to use it as y to the one-third. All right if you go back and rearrange the original equation and solve for x you get y to the one-third. All right so those become the pieces that we're going to need to set up our integral. So let's go back to the previous page and we'll go ahead and set it up. So our volume since it was a dy problem our limits of integration need to be y values so let's take a look at the graph again and our y values are going from zero on the low end up to a positive one and pi then remember we need big r squared so big r remember was one minus little r was one minus y to the one-third and just don't forget to square that. All right so you are most welcome to evaluate that in your calculator. You could just do the definite integral part excluding the pi if you wanted to do that you would get 0.9 pi interesting it comes out like that it's actually 0.9 if you wanted to multiply the pi in and just obtain a pure decimal answer it would be approximately 2.827 and again if by chance you needed to include units of measure since it is a volume problem it would be in cubic units