Using Integrals to Solve Applied Problems

If a curve is rotated about the x or y axis or a line parallel to one of the axes, to form a 3D object, we can calculate the volume of this solid by using the tools of integration. For example, let's say the parabola y = x2 is rotated around its axis of symmetry to form a paraboloid, as in Figure 10.16:

Figure 10.16: A parabola rotated about the z axis

We can find the volume by adding up all the slices of the paraboloid as you go up the solid. Just as before, when we were using rectangles in two dimensions, now we're using cylinders in three dimensions. In Figure 10.16, the slices are going up the figure and not to the right, so we can flip it in our heads and redefine the curve y = x2 as y = sqrt(x).

Now the radius of each cylinder is the y value, and let's say we're going from x = 0 to x = 1:

Figure 10.17: Flipping the paraboloid on its side

The endpoints are still 0...