Evaluate the integral.
-$$\int _ { 4 } ^ { 8 } \int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \pi / 2 } ( \varrho \sin \varphi ) ^ { 2 } \mathrm {~d} \theta \mathrm { d } \varphi \mathrm { d \varphi }$$
A) $\frac { 224 } { 9 } \pi ^ { 2 }$
B) $\frac { 56 } { 3 } \pi ^ { 2 }$
C) $\frac { 56 } { 3 } \pi$
D) $\frac { 224 } { 9 } \pi$

Question

Quizplus · Accepted Answer

The integral can be separated into three parts due to its multiplicative nature. The integral over $\theta$ from $0$ to $\frac{\pi}{2}$ is simply $\frac{\pi}{2}$, as it does not depend on $\theta$. The integral over $\varphi$ involves $\sin^2(\varphi)$, which averages to $\frac{1}{2}$ over a period, thus giving $\frac{\pi}{4}$ for the bounds from $0$ to $\frac{\pi}{2}$. The integral over $\varrho$ from $4$ to $8$ is $\frac{\varrho^3}{3}$ evaluated from $4$ to $8$, resulting in $\frac{512}{3} - \frac{64}{3} = \frac{448}{3}$. Multiplying these results together gives $\frac{\pi}{2} \times \frac{\pi}{4} \times \frac{448}{3} = \frac{56}{3} \pi^2$.

Evaluate the Integral A) 2249π2\frac { 224 } { 9 } \pi ^ { 2 }9224​π2

Evaluate the Integral A) $\frac { 224 } { 9 } \pi ^ { 2 }$