Find the volume of the described solid.
-The solid lies between planes perpendicular to the $x$-axis at $x = - 6$ and $x = 6$. The cross sections perpendicular to the $x$-axis are semicircles whose diameters run from $y = - \sqrt { 36 - x ^ { 2 } }$ to $y = \sqrt { 36 - x ^ { 2 } }$.
A) $144 \pi$
B) $288 \pi$
C) $72 \pi$
D) $576 \pi$

Question

Quizplus · Accepted Answer

The volume of the solid can be calculated by integrating the area of the cross-sections along the x-axis. The diameter of the semicircles is $2\sqrt{36-x^2}$, so the radius is $\sqrt{36-x^2}$. The area of a semicircle is $\frac{1}{2}\pi r^2$, so the area of each cross-section is $\frac{1}{2}\pi(\sqrt{36-x^2})^2 = \frac{1}{2}\pi(36-x^2)$. Integrating this from $-6$ to $6$ gives the volume: $\int_{-6}^{6} \frac{1}{2}\pi(36-x^2) dx = \frac{1}{2}\pi [36x-\frac{x^3}{3}]_{-6}^{6} = \frac{1}{2}\pi [216-\frac{216}{3}] = 144\pi$.

Find the Volume of the Described Solid xxx -Axis At x=−6x = - 6x=−6

Find the Volume of the Described Solid $x$ -Axis At $x = - 6$