Use spherical coordinates to evaluate $\iiint _ { B } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V$ where B is the ball $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 8$
A) 512 $ \pi$
B) 8 $ \pi$
C) 64 $ \pi$
D) 1024 $ \pi$

Question

Quizplus · Accepted Answer

The integral represents the moment of inertia of a solid ball of radius $\sqrt{8}$ about its center. In spherical coordinates, the integrand $\sqrt{x^2 + y^2 + z^2}$ becomes $ \rho $, and the volume element $dV$ becomes $\rho^2 \sin \phi \, d\rho \, d\phi \, d\theta$. The limits for $\rho$ are from 0 to $\sqrt{8}$, for $\phi$ from 0 to $\pi$, and for $\theta$ from 0 to $2\pi$. The integral evaluates to $64 \pi$.

Use Spherical Coordinates to Evaluate ∭Bx2+y2+z2dV\iiint _ { B } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V∭B​x2+y2+z2​dV

Use Spherical Coordinates to Evaluate $\iiint _ { B } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V$