Evaluate the iterated integral $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { 1 } \rho ^ { 2 } \sin \phi d \rho d \phi d \theta$ .
A) $\frac { \pi } { 6 }$

B) $\frac { \pi } { 4 }$

C)4$\pi$

D)2$\pi$

E) $\frac { \pi } { 16 }$

F) $\frac { \pi } { 2 }$

G) $\frac { \pi } { 3 }$

H) $\frac { 4 \pi } { 3 }$

Question

Evaluate the iterated integral $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { 1 } \rho ^ { 2 } \sin \phi d \rho d \phi d \theta$ .
A) $\frac { \pi } { 6 }$

B) $\frac { \pi } { 4 }$

C)4$\pi$

D)2$\pi$

E) $\frac { \pi } { 16 }$

F) $\frac { \pi } { 2 }$

G) $\frac { \pi } { 3 }$

H) $\frac { 4 \pi } { 3 }$

Quizplus · Accepted Answer

The integral can be evaluated by integrating step by step. First, integrate $\rho^2$ with respect to $\rho$ from 0 to 1, which gives $\frac{\rho^3}{3}$ evaluated from 0 to 1, resulting in $\frac{1}{3}$. Then, integrate $\sin\phi$ with respect to $\phi$ from 0 to $\frac{\pi}{2}$, which gives $-\cos\phi$ evaluated from 0 to $\frac{\pi}{2}$, resulting in 1. Finally, integrate 1 with respect to $\theta$ from 0 to $\frac{\pi}{2}$, which gives $\theta$ evaluated from 0 to $\frac{\pi}{2}$, resulting in $\frac{\pi}{2}$. Multiplying these results gives $\frac{1}{3} \times 1 \times \frac{\pi}{2} = \frac{\pi}{6}$.

Evaluate the Iterated Integral ∫0π/2∫0π/2∫01ρ2sin⁡ϕdρdϕdθ\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { 1 } \rho ^ { 2 } \sin \phi d \rho d \phi d \theta∫0π/2​∫0π/2​∫01​ρ2sinϕdρdϕdθ

Evaluate the Iterated Integral $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { 1 } \rho ^ { 2 } \sin \phi d \rho d \phi d \theta$