Use a spherical coordinate integral to find the volume of the given solid.
-the solid bounded below by the hemisphere $\varrho = 1 , z \geq 0$, and above by the cardioid of revolution $\varrho = 9 + 5$ cos $\varphi$
A) $\frac { 6367 } { 6 } \pi$
B) $\frac { 6367 } { 10 } \pi$
C) $\frac { 6367 } { 8 } \pi$
D) $\frac { 6367 } { 9 } \pi$

Question

Quizplus · Accepted Answer

The volume of the solid can be found using the spherical coordinate integral formula for volume, $V = \int \int \int_{E} \varrho^2 \sin \varphi \, d\varrho \, d\varphi \, d\theta$, where $E$ is the region bounded by the given surfaces. The lower bound is the hemisphere $\varrho = 1$, and the upper bound is the cardioid of revolution $\varrho = 9 + 5\cos\varphi$. Integrating over the appropriate limits for $\varrho$, $\varphi$, and $\theta$, and performing the necessary calculations yields the volume $\frac{6367}{6}\pi$.

Use a Spherical Coordinate Integral to Find the Volume of the Given