 Hi everyone, this will be an example for you of another disk method problem to find the volume of a solid of revolution. So here we have a region bounded by the curve y equals x squared minus x, the x-axis, and between the two vertical lines x equals negative one and x equals positive one. And we are revolving it around the x-axis. So go ahead and graph those in your calculator. You should get a picture that looks like this. And I did adjust my window so you could see this a little bit better. Let's go ahead and draw in those vertical lines at negative one and positive one. And our axis of revolution is the x-axis. So the regions, really we have two of them in this case since we're in between the two vertical lines x equals negative one and positive one. The regions we are revolving around that x-axis are the ones that you see in yellow. So if we were to draw in our representative rectangles into each of those yellow regions, remember since we are talking about the disk method, two key characteristics are that our representative rectangles must be perpendicular to the axis of revolution and also touching it. Now remember, again, this representative rectangle really could have been drawn anywhere. I'm just choosing to draw it where you see I did. So imagine taking each of those rectangles and revolving it around the x-axis. So we really are going to need two integrals to evaluate and find their sum in order to find the total volume of the solid that is generated. The radius we're going to need, in this case, we'll need two of them, are the two different lengths of our rectangles. So let's go ahead and set that up on the previous page. So the first rectangle, the one that was over on the left, I'll call that R1, remember that in order to find the length of that rectangle, it's going to be top minus bottom. So at the top it's hitting the curve, x squared minus x, and on the bottom it's obviously hitting the x-axis. The other rectangle, again, we can think of it doing top minus bottom. This time the top is hitting the x-axis, which would be y equals 0, and the bottom is hitting the curve. So what we end up with there is 0 minus the quantity x squared minus x. So when we go to find our volume, so the first integral is going to go from negative 1 to 0, and that's going to be of the quantity x squared minus x, and we need to square that according to the formula. And to that we're going to add the other integral that goes 0 to 1. I forgot my pi. So you can evaluate each one separately on your calculator. You can either answer in terms of pi, if you do, you should get 1.06 repeating. That would be just the definite integrals themselves without the pi, and then you can multiply that by pi. If you want a pure decimal answer, you should be getting approximately 3.351. If by chance you needed to have units, remember these are volume problems, so they would be cubic units on your answer.