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

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:
$\int_{0}^{1} 2 π x f (x) d x = \int_{0}^{b} π (f^{- 1} (y))^{2} d y .$
Substitute $y = f (x)$ in the second. Also substitute $d y = f^{'} (x) d x$ . Integrate by parts to reach the first.

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

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