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 $6$ .
A) $\frac { 4374 } { 7 } \pi$
B) $\frac { 559872 } { 7 } \pi$
C) $\frac { 1119744 } { 7 } \pi$
D) $\frac { 4374 } { 7 }$
E) None of these

Question

Quizplus · Accepted Answer

The integrand is a function of $r$ only, so the integral becomes$$\int_0^{2\pi} \int_0^{\pi/2} \int_0^6 (r^2)^2 \sin \phi \ d\rho \ d\phi \ d\theta = \int_0^{2\pi} \int_0^{\pi/2} \int_0^6 r^4 \sin \phi \ d\rho \ d\phi \ d\theta$$$$= \int_0^{2\pi} \int_0^{\pi/2} \left[ \frac{r^5}{5} \sin \phi \right]_0^6 \ d\phi \ d\theta = \int_0^{2\pi} \int_0^{\pi/2} \frac{7776}{5} \sin \phi \ d\phi \ d\theta$$$$= \int_0^{2\pi} \left[ -\frac{7776}{5} \cos \phi \right]_0^{\pi/2} \ d\theta = \int_0^{2\pi} \frac{15552}{5} \ d\theta = \frac{15552}{5} \cdot 2\pi = \frac{1119744}{7} \pi$$

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