Use the Theorem of Pappus to find the volume of the solid obtained by revolving the region bounded by the graphs of $y = 16 - x ^ { 2 } , y = 16$, and $x = 4$ about the $y$-axis.
A) $128 \pi$
B) $\frac { 512 } { 3 } \pi$
C) $\frac { 128 } { 3 } \pi$
D) $32 \pi$

Question

Quizplus · Accepted Answer

The volume of the solid is given by the integral of the area of the cross-sections perpendicular to the $y$-axis. The area of each cross-section is the area of a circle with radius $8 - \frac{1}{2}x$. Therefore, the volume of the solid is$$V = \int_0^4 \pi (8 - \frac{1}{2}x)^2 dx = \pi \int_0^4 (64 - 16x + \frac{1}{4}x^2) dx = \pi \left[64x - 8x^2 + \frac{1}{12}x^3\right]_0^4 = \pi \left(128 - 128 + \frac{64}{3}\right) = \frac{128}{3}\pi.$$

Use the Theorem of Pappus to Find the Volume of the Solid