Equivalence of the washer and shell methods for finding volume Let $f$ be differentiable and increasing on the interval $a \leq x \leq b$ with $a>0,$ and suppose that $f$ has a differentiable inverse, $f^{-1}$ Revolve about the $y$ -axis the region bounded by the graph of $f$ and the lines $x=a$ and $y=f(b)$ to generate a solid. Then the values of the integrals given by the washer and shell methods for the volume have identical values:
$$\int_{f(a)}^{f(b)} \pi\left(\left(f^{-1}(y)\right)^{2}-a^{2}\right) d y=\int_{a}^{b} 2 \pi x(f(b)-f(x)) d x$$
To prove this equality, define
$$W(t)=\int_{f(a)}^{f(t)} \pi\left(\left(f^{-1}(y)\right)^{2}-a^{2}\right) d y$$
$$S(t)=\int_{a}^{t} 2 \pi x(f(t)-f(x)) d x$$
Then show that the functions $W$ and $S$ agree at a point of $[a, b]$ and have identical derivatives on $[a, b] .$ As you saw in Section 4.7 Exercise $90,$ this will guarantee $W(t)=S(t)$ for all $t$ in $[a, b] .$ In particular, $W(b)=S(b) .$ (Source: "Disks and Shells Revisited," by Walter Carlip, American Mathematical Monthly, Vol. $98,$ No. 2 Feb. $1991,$ pp. $154-156 .$ )