Chapter 6.3, Problem 164E

Consider the function $y=f\left(x\right)$ , which decreases from $f\left(0\right)=b$ to $f\left(1\right)=0$ . Set up the integrals for determining the volume, using both the shell method and the disk method, of the solid generated when this region, with $x=0$ and $y=0$ , is rotated around the y-axis.

Prove that both methods approximate the same volume. Which method is easier to apply? (Hint: Since f(x) is one- to-one, there exists an inverse $f^{-1}\left(y\right)$ ).