Area and Volume Revisited — Question 7

Question 7 – Solution

Strategy. Although the slices have different shapes, compare their areas at the same height.

Step 1: Slice areas

At height $z$, the slice of $S_1$ is a disk of radius $z$, so $$A_1(z)=\pi z^2.$$ The slice of $S_2$ is a rectangle of side lengths $z^2$ and $\pi$, so $$A_2(z)=z^2\pi=\pi z^2.$$

Step 2: Compare and integrate Both solids occupy $0\le z\le 1$ and have equal cross-sectional area at every height. Cavalieri’s principle therefore gives equal volumes, and $$V_1=V_2=\int_0^1\pi z^2\,dz =\pi\left[\frac{z^3}{3}\right]_0^1 =\boxed{\frac{\pi}{3}}.$$

Verification The conclusion depends on slice areas, not slice shapes: circular slices in $S_1$ and rectangular slices in $S_2$ accumulate the same volume height by height.