The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.

Region Between Two Curves - Solid of Revolution

Now we consider the region R between two curves y = f(x) and y = g(x) from x = a to x = b. Rotating R about the x-axis generates a solid of revolution S shown in Figure 6.2.6(c).


Figure 6.2.6 (a): R2, S2


Figure 6.2.6 (b): R1, S1



Figure 6.2.6 (c) R = R2 - R1, S = S2 - S1

Let R1 be the region under the curve y = f(x) shown in Figure 6.2.6(b), and R2, the region under the curve y = g(x), shown in Figure 6.2.6(a). Then S can be found by removing the solid of revolution S2 generated by R1 from the solid of revolution S2 generated by R2. Therefore

volume of S = volume of S2 - volume of S1.

This justifies the formula


We combine this into a single integral.

VOLUME BY DISC METHOD 06_applications_of_the_integral-38.gif

Another way to see this formula is to divide the solid into annular discs (washers) with inner radius f(x) and outer radius g(x), as illustrated in Figure 6.2.7.


Figure 6.2.7

Example 2

Last Update: 2006-11-22