 Hi everyone, we are going to take a look at an example of finding the volume of a solid of revolution by using the disk method. Now this one's going to end up being a dy problem as opposed to a dx problem. So if you take a look at the problem, we are trying to find the volume of the solid generated by revolving about the line x equals 1, so that's our axis of revolution. The region bounded by the curve x minus 1 the quantity squared equals 20 minus 4y, also bounded by the lines x equals 1, y equals 1, and y equals 3 to the right of x equals 1. That's a lot to take in. So you're going to want to graph this on your calculator. Obviously that equation is going to be requiring a little bit of rearranging to get into your calculator. So probably the best way, let's see if we subtract the 20 from both sides, divide both sides by the negative 4, that will give us something to easily enter in as our y equation. So if you go ahead and do that, you can also enter in as y2 and y3, the lines y equals 1 and y equals 3. So when you get that set up and I admit I did change my window a little bit, hopefully you have something that looks like this, the blue curve. That is the curve. That's the quadratic, obviously, and you can see in red the line y equals 1 and in black the line y equals 3. So what we're trying to do, remember there were a couple vertical lines here mentioned, so we had the vertical line at x equals 1. So let's go ahead and draw that in. That is our axis of revolution, remember. And we want to revolve the region to the right of that bounded by the curves. So it ends up that it is the yellow region that we want to revolve. Oftentimes the hardest part of these problems is just making sure you're reading them correctly to ensure that you have the correct region of revolution. So this is the one we're talking about. So we want to take that yellow region and flip it around that vertical axis of revolution at x equals 1. So of course we are studying the disc method and two major characteristics of the representative rectangle that we will draw to represent the disc method is that that rectangle needs to be perpendicular to the axis of revolution and also touching it. Well, the only way that's going to happen is if our representative rectangle was to go this way and because of its orientation, that's why it becomes a dy problem. So imagine taking that representative rectangle and flipping it around the axis so it goes over to that side. So imagine a whole lot of like flat frisbees. So if you think of the formula that we need to use, let's go ahead and write it out. According to a disc method, the volume of course is going to be an integral from a to b. Now remember our limits of integration because this is the dy problem will have to be y values of pi and the radius squared. Now because of the way our representative rectangle is oriented, our radius is essentially the length of the rectangle from left to right. So we need to do the right minus left in order to get the length of that rectangle. So on the right side, it's hitting the parabola. So let's go ahead and start writing this out. So our r value, our radius, on the right side, it's hitting the quantity square root of 20 minus 4y plus 1. Now you might ask where that came from for the right side. Remember this is a dy problem. So we had to take that original equation and express it in terms of x in terms of y. So even though we had to rearrange it to get it into our calculator by solving for y, now you need to solve for x. So going back to the original equation, take the square root of both sides, and then you need to add the 1 over. So this becomes what it's hitting on the right side. Minus. Now on the left side, it's simply hitting, if you go back to your picture, the representative rectangle on the left side is hitting that vertical line at x equals 1. So this is simply going to be minus 1. So obviously the ones cancel out and all you're left with is the square root quantity. So that is the length of our representative rectangle. So let's go ahead and set up our volume then. Say we will have that the volume is, now remember our limits of integration have to be y values. So in this case, they are 1 to 3 of pi and we need to take that r quantity and square it. Obviously the square root and the square cancel each other out. From there you can evaluate it in your calculator or do it by hand. This one's not so bad by hand since the square and the square root cancel each other out. If you care to do an exact answer and get it in terms of pi, it would be 24 pi. If you want a pure decimal answer rounded to the nearest thousandth, again you can do this on your calculator or by hand. You should get approximately 75.398.