 A useful idea for two-dimensional objects—tape and scissors—we can form a geometric figure by taping pieces together and cutting off parts we don't want. Three-dimensional objects are no different. For example, say we want to find the volume of the object formed by the region between y equals square root of x and y equals x is revolved around the x-axis. That gives us a figure like this, which is sort of a cup. Now if you were going to make a cup out of wood, what you would do is you'd start with a block of wood and get rid of the parts that you don't want. But this is calculus, so instead we'll start with a larger piece and get rid of the parts we don't want. And one way we can do this is we can fill in the area between the region and the axis of rotation, revolve the new region around the x-axis, then remove the inside part which leaves the volume that we want. Now let's find that volume. The two curves intersect at 0, 0 and 1, 1. Now if we fill in the part between the region and the x-axis, then revolve around the x-axis. The volume of the solid produced from the whole region is revolved around the x-axis, is going to be, but we want to remove the inside. So we need to figure out what the volume of the inside figure is. So remember we filled in our region, but we will eventually remove this fill-in part. So let's figure out what that volume is. If we just consider the volume of the solid formed by revolving the fill-in part around the x-axis, that volume will be, and so the volume will be the whole figure minus the inside figure, and we can compute the value and find our volume.