 Oh, and welcome to this quick recap of section 6.2, using definite integrals to find volume. The big idea that we'll keep returning to in this section is this. If we have a three-dimensional shape, such as this cone, we can imagine slicing it up into many thin pieces that have a nice regular shape. For example, slicing it up into many thin cylinders. This can be thought of in the same way as when we approximated the area under a curve by using rectangles. Now we're approximating the volume of shape using cylinders. We can look at a side view of this with one of these cylinders drawn in. We'll call this a representative slice. If we can figure out the volume of the cylinder, that can help us figure out the volume of the entire shape. Well, the cylinder has width delta x, that's a very thin slice in the x direction. It also has a radius that we know. If we know a formula for this blue line, f of x, then that means we also know the radius, and it's the same thing, f of x. That lets us write the volume of a slice, and this is just the area of a cylinder, pi times a radius squared times delta x. Now we want to add all of these slices up and take the limit as a number of slices goes to infinity. This is the same process as adding up all of the boxes representing the area under a curve and taking the limit as x goes to infinity. That gives us an integral. Here this integral represents adding up all of these cylinders, and you can see the formula in this integral for the volume of a cylinder. We'll take this idea to the idea of a volume or a solid of revolution. This is a three-dimensional solid generated by revolving a two-dimensional region around a fixed axis. Here we're going to revolve this orange colored region around the x axis. Along the way, we're going to follow the path of a thin box with width delta x. The height of this box is r of x given by the function that we were initially given. If we think of that two-dimensional region as sitting inside three-dimensional space, we can follow it as we revolve this region. Here we revolve it, and a bit more. Until we're all the way around, we can see that the box has turned into a cylinder, and the region has turned into an interesting three-dimensional bowl shape. Here's a solid view of the same shape. Here we have the region and the box, and the resulting cylinder inside the three-dimensional shape. Because the box has height r of x, this corresponds to the radius of the cylinder once we've revolved it. Similarly, because the box has width delta x, the cylinder has height delta x. That's everything we need in order to write the volume of a cylinder, and we can see how it shows up in this integral here. We have pi times a radius squared times delta x. When we take the limit as the number of cylinders goes to infinity, we get an integral, and that integral adds up the volumes of all these cylinders, giving us the exact volume of the shape, in the same way that an integral gave us the exact area under a curve. This is called the disk method because the cylinder shape can also be called a disk, like the shape of a coin or a poker chip. Sometimes the region that we're revolving doesn't touch the x-axis, and this in turn gives us a shape called a washer. A washer is really the same thing as a cylinder with a hole drilled out of the center. We're going to think of this as one cylinder with another cylinder subtracted from it. You can see that in the formula here. In red we have pi capital r of x squared, that's the outer edge of the cylinder. And on the right-hand side in the three-dimensional region, we can see how that same radius, capital R, gives us the outside radius of a washer. In blue we have the inner radius of the inside edge, and in this integral we're subtracting the area of a cylinder that sits inside the larger cylinder. This formula for washers can be useful if your region doesn't touch the x-axis. We can also revolve around the y-axis, and we might get disks or washers depending on whether the region touches the y-axis or not. Take a moment and see if you can figure out what the volume of this region should be, where the outer radius is capital R of y, and the inner radius is little r of y. Here again is that formula, and we can see that it is once again in the form the area of one cylinder minus the area of a smaller inside cylinder. Now that we've seen these examples, let's take a look at some concrete examples of these. One thing in particular to note is that visualizing these shapes takes practice, but that's a trainable skill, and we'll take a look at many more of these examples in the videos to come.