Suppose … decreases from … to

Suppose $y=f(x)$ decreases from $f(0)=b$ to $f(1)=0$. The curve is rotated around the $y$ axis. Compare shells to disks:
$${\int }_0^{1}2\pi xf(x)dx={\int }_0^b\pi (f^{-1}(y){)}^{2}dy.$$
Substitute $y=f(x)$ in the second. Also substitute $dy=f^{\prime }(x)dx$. Integrate by parts to reach the first.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §8.1 Areas and Volumes by Slices, #52