Use spherical coordinates to evaluate $\iiint _ { B } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V$, where $B$ is the ball $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 10$.
A) $10000 \pi$
B) $10 \pi$
C) $1000 \pi$
D) $2000 \pi$

Question

Quizplus · Accepted Answer

The volume element in spherical coordinates is $dV = \rho^2 \sin \phi d\rho d\phi d\theta$. So,$$\begin{split} \iiint_B \sqrt{x^2 + y^2 + z^2} dV &= \int_0^{2\pi} \int_0^\pi \int_0^1 \rho^2 \sin \phi d\rho d\phi d\theta \\ & = \int_0^{2\pi} \int_0^\pi \left[ \frac{\rho^3}{3} \right]_0^1 \sin \phi d\phi d\theta \\ & = \int_0^{2\pi} \int_0^\pi \frac{1}{3} \sin \phi d\phi d\theta \\ & = \int_0^{2\pi} \left[ -\frac{1}{3} \cos \phi \right]_0^\pi d\theta \\ & = \int_0^{2\pi} \frac{2}{3} d\theta \\ & = \frac{4\pi}{3}. \end{split}$$

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 $\iiint _ { B } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V$