Use spherical coordinates.
Evaluate $\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V$, where $B$ is the ball with center the origin and radius 4 .
A) $\frac { 65536 } { 7 } \pi$
B) $\frac { 4374 } { 7 }$
C) $\frac { 4374 } { 7 } \pi$
D) $\frac { 559872 } { 7 } \pi$
E) None of these

Question

Quizplus · Accepted Answer

In spherical coordinates, $x^2 + y^2 + z^2 = \rho^2$, where $\rho$ is the radial distance from the origin. Thus, the integral becomes $\iiint_B \rho^4 dV$. The volume element in spherical coordinates is $dV = \rho^2 \sin \phi d\rho d\theta d\phi$, where $0 \leq \rho \leq 4$, $0 \leq \theta \leq 2\pi$, and $0 \leq \phi \leq \pi$. Integrating $\rho^4 \rho^2 \sin \phi$ over these limits gives $\frac{65536}{7}\pi$.

Use Spherical Coordinates ∭B(x2+y2+z2)2dV\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V∭B​(x2+y2+z2)2dV

Use Spherical Coordinates $\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V$