Use the Theorem of Pappus to find the volume of the solid obtained by revolving the region bounded by the graphs of $y=36-x^{2}, y=36$ and $x=6$ about the y-axis.
A) $648$ $\pi$
B) $144$ $\pi$
C) $864$ $\pi$
D) $108$ $\pi$

Question

Quizplus · Accepted Answer

The volume of the solid is given by the integral of the area of the cross-sections of the solid. The cross-sections are circles with radius $r=36-x^{2}$. The area of each cross-section is $\pi r^{2}=\pi(36-x^{2})^{2}$. The limits of integration are $x=0$ and $x=6$. Therefore, the volume of the solid is$$V=\int_{0}^{6}\pi(36-x^{2})^{2}dx=\pi\int_{0}^{6}(1296-72x^{2}+x^{4})dx=\pi\left[1296x-24x^{3}+\frac{1}{5}x^{5}\right]_{0}^{6}=648\pi$$

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