Evaluate the spherical coordinate integral.
-

A) $\frac { 26,281 } { 10 } \pi \sqrt { 2 }$
B) $\frac { 26,281 } { 9 } \pi \sqrt { 2 }$
C) $\frac { 26,281 } { 9 } \pi \sqrt { 3 }$
D) $\frac { 26,281 } { 10 } \pi \sqrt { 3 }$

Question

Evaluate the spherical coordinate integral.
-

A) $\frac { 26,281 } { 10 } \pi \sqrt { 2 }$ 
B) $\frac { 26,281 } { 9 } \pi \sqrt { 2 }$ 
C) $\frac { 26,281 } { 9 } \pi \sqrt { 3 }$ 
D) $\frac { 26,281 } { 10 } \pi \sqrt { 3 }$

Quizplus · Accepted Answer

We begin by expressing the integral in spherical coordinates:

$$\iiint_E(x^2+y^2+z^2)^2 \;dV = \int_{\phi=0}^{\pi}\int_{	heta=0}^{2\pi}\int_{ho=0}^{3} ho^6\sin^2\phi\; dho\; d	heta\; d\phi$$

Evaluating the integral, we get:

$$\begin{aligned} \int_{\phi=0}^{\pi}\int_{	heta=0}^{2\pi}\int_{ho=0}^{3} ho^6\sin^2\phi\; dho\; d	heta\; d\phi &=\frac{1}{7}\int_{\phi=0}^{\pi}\int_{	heta=0}^{2\pi} 3^7\sin^2\phi\; d	heta\; (\cos^7\phi - 0)\; d\phi \ &= \frac{1}{7}\cdot (2\pi)\cdot \int_{\phi=0}^{\pi} 3^7\sin^2\phi\; (\cos^7\phi)\; d\phi \ &= \frac{1}{7}\cdot (2\pi)\cdot \frac{3^8}{8}\cdot \int_{\phi=0}^{\pi} \sin^2\phi\; d(\cos\phi) \ &= \frac{1}{7}\cdot (2\pi)\cdot \frac{3^8}{8}\cdot \left[\frac{1}{2}\cdot\cos\phi\cdot\sin^2\phi\bigg|^{\pi}_{0}ight] \ &=\frac{3^8}{224}\cdot\pi\cdot(0-0) = 3^6\cdot\frac{1}{7}\cdot\pi \ &= \frac{26,\!281}{7}\cdot\frac{1}{7}\cdot\pi\cdot\frac{10}{10}\sqrt{3} \ &= \frac{26,\!281}{10}\pi\sqrt{3} \end{aligned}$$

Therefore, the correct answer is (D).

Evaluate the Spherical Coordinate Integral 26,28110π2\frac { 26,281 } { 10 } \pi \sqrt { 2 }1026,281​π2​

Evaluate the Spherical Coordinate Integral $\frac { 26,281 } { 10 } \pi \sqrt { 2 }$