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 integral in spherical coordinates is$$\iiint_B \sqrt{x^2+y^2+z^2} dV = \int_0^{2\pi} \int_0^\pi \int_0^2 ho^2 \sin \phi dho d\phi d	heta = \int_0^{2\pi} \int_0^\pi \left[ \frac{ho^3}{3} ight]_0^2 \sin \phi d\phi d	heta = \int_0^{2\pi} \int_0^\pi \frac{8}{3} \sin \phi d\phi d	heta = \int_0^{2\pi} \left[ -\frac{8}{3} \cos \phi ight]_0^\pi d	heta = \int_0^{2\pi} \frac{16}{3} d	heta = \frac{32\pi}{3} = 10000 \pi.$$

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