Use a spherical coordinate integral to find the volume of the given solid.
-the solid enclosed by the cardioid of revolution $\varrho = 5 + 3 \cos \varphi$
A) $\frac { 340 } { 3 } \pi$
B) $\frac { 680 } { 3 } \pi$
C) $\frac { 1360 } { 3 } \pi$
D) $\frac { 170 } { 3 } \pi$

Question

Quizplus · Accepted Answer

The volume of the solid can be found by integrating in spherical coordinates. The limits for $\varphi$ are from 0 to $\pi$, and for $\theta$ from 0 to $2\pi$. The radial coordinate $\varrho$ varies from 0 to $5 + 3 \cos \varphi$. The volume integral is $\int_0^{2\pi} \int_0^{\pi} \int_0^{5+3\cos\varphi} \varrho^2 \sin\varphi \, d\varrho \, d\varphi \, d\theta$, which evaluates to $\frac{680}{3} \pi$.

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