Use spherical coordinates to evaluate $\iiint _ { B } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V$, where $B$ is the ball $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 7$.
Select the correct answer.
A) $2401 \pi$
B) $343 \pi$
C) $686 \pi$
D) $7 \pi$

Question

Quizplus · Accepted Answer

The volume element in spherical coordinates is $dV = \rho^2 \sin \phi d\rho d\phi d\theta$. So,$$\iiint_B \sqrt{x^2 + y^2 + z^2} dV = \int_0^{2\pi} \int_0^{\pi} \int_0^7 \rho^2 \sin \phi d\rho d\phi d\theta$$$$= \int_0^{2\pi} \int_0^{\pi} \left[\frac{\rho^3}{3}\right]_0^7 \sin \phi d\phi d\theta$$$$= \int_0^{2\pi} \int_0^{\pi} \frac{343}{3} \sin \phi d\phi d\theta$$$$= \int_0^{2\pi} \left[-\frac{343}{3} \cos \phi\right]_0^{\pi} d\theta$$$$= \int_0^{2\pi} \frac{686}{3} d\theta$$$$= \frac{686}{3} \cdot 2\pi$$$$= 2401 \pi$$

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