Use the Theorem of Pappus to find the volume of the solid obtained by revolving the region bounded by the graphs of $y = 4 - x ^ { 2 } , y = 4$, and $x = 2$ about the $y$-axis. Select the correct answer.
A) $8 \pi$
B) $\frac { 32 } { 3 } \pi$
C) $4 \pi$
D) $\frac { 16 } { 3 } \pi$

Question

Quizplus · Accepted Answer

The volume of the solid obtained by revolving the region bounded by the graphs of $y = 4 - x ^ { 2 } , y = 4$, and $x = 2$ about the $y$-axis is given by the Theorem of Pappus as$$V = 2\pi \int_0^4 x (4 - x^2) \ dx = 2\pi \left[ 4x - \frac{x^3}{3} \right]_0^4 = 2\pi \left( 16 - \frac{64}{3} \right) = 2\pi \cdot \frac{32}{3} = \frac{64\pi}{3}.$$

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