13. Volume

f1. Strategy for Computing Volumes of Revolution

When we need to compute the volume of a solid of revolution, we will need to decide whether to do an \(x\) integral or a \(y\) integral and whether to use disks, washers or cylinders. Do not simply try to remember all the formulas for the appropriate cases. Rather, here are several steps to follow to help make these decisions. We suggest you print out a copy of this page so you can refer to it as you do the problems.

1. Draw the region in the \(xy\)-plane.
2. Decide if we will do an \(x\)-integral or a \(y\)-integral.

This decision should be made in exactly the same way we would decide how to compute the area of the region, without regard to the axis of rotation. Is it easier to compute the area as an \(x\) integral or as a \(y\) integral? Do the same for the volume.

3. Determine the limits for your integral.

Do this exactly as we would for computing the area.

4. Draw one Riemann sum rectangle in the figure.

For an \(x\)-integral the rectangle is vertical. For a \(y\)-integral the rectangle is horizontal.

5. Rotate the rectangle about the specified axis and observe if it sweeps out a disk, a washer or a cylinder.
6. Express everything in terms of the independent variable \(x\) or \(y\).
   1. For a disk, determine the radius.
   2. For a washer, determine the inner radius and outer radius.
   3. For a cylinder, determine the radius and height.
7. Compute the volume integral:
   1. For a disk, \(\displaystyle V=\int_a^b \pi(\text{radius})^2\,dx\)   or   \(\displaystyle V=\int_a^b \pi(\text{radius})^2\,dy\).
   2. For a washer, \(\displaystyle V=\int_a^b \pi(\text{upper})^2-\pi(\text{lower})^2\,dx\)   or   \(\displaystyle V=\int_a^b \pi(\text{right})^2-\pi(\text{left})^2\,dy\).
   3. For a cylinder, \(\displaystyle V=\int_a^b 2\pi(\text{radius})(\text{height})\,dx\)   or   \(\displaystyle V=\int_a^b 2\pi(\text{radius})(\text{height})\,dy\).

Be sure to do the exercises for this section, because they don't tell you which method (disks, washers or cylinders) to use.