Find the volume of the described solid.
-The solid lies between planes perpendicular to the $x$-axis at $x = - 3$ and $x = 3$. The cross sections perpendicular to the $x$-axis are semicircles whose diameters run from $y = - \sqrt { 9 - x ^ { 2 } }$ to $y = \sqrt { 9 - x ^ { 2 } }$.
A) $9 \pi$
B) $36 \pi$
C) $72 \pi$
D) $18 \pi$

Question

Quizplus · Accepted Answer

The volume of the solid can be calculated using the formula for the volume of a solid of revolution, specifically using the method of disks or washers. Here, the cross-sections are semicircles with diameters from $y = -\sqrt{9 - x^2}$ to $y = \sqrt{9 - x^2}$, which means the radius of each semicircle is $\frac{\sqrt{9 - x^2}}{2}$. The area of each semicircle is $\frac{1}{2} \pi r^2 = \frac{1}{2} \pi \left(\frac{\sqrt{9 - x^2}}{2}\right)^2$. Integrating this area from $x = -3$ to $x = 3$ gives the volume of the solid. The integral simplifies to $\int_{-3}^{3} \frac{1}{2} \pi \left(\frac{9 - x^2}{4}\right) dx$, which evaluates to $18 \pi$.

Find the Volume of the Described Solid xxx -Axis At x=−3x = - 3x=−3

Find the Volume of the Described Solid $x$ -Axis At $x = - 3$