Evaluate the following iterated integral.
$ \int_{0}^{\pi /15}\int_{0}^{\pi/15}\int_{0}^{cos\theta}\rho^{2} \sin \phi \cos \phi d \rho d \theta d \phi $
A) $\frac { 1 } { 36 } \left( 5 + \cos \frac { 2 \pi } { 15 } \right) \sin ^ { 3 } \frac { \pi } { 15 }$
B) $\frac { 1 } { 6 } \left( 5 + \cos \frac { 2 \pi } { 15 } \right) \sin ^ { 3 } \frac { \pi } { 15 }$
C) $\frac { 1 } { 6 } \left( 3 + \cos \frac { 2 \pi } { 15 } \right) \sin ^ { 2 } \frac { \pi } { 15 }$
D) $\frac { 1 } { 36 } \left( 5 + \cos \frac { \pi } { 15 } \right) \sin ^ { 3 } \frac { \pi } { 15 }$
E) $\frac { 1 } { 36 } \left( 5 + \cos \frac { 2 \pi } { 15 } \right) \sin ^ { 2 } \frac { \pi } { 15 }$

Question

Evaluate the following iterated integral. 
$ \int_{0}^{\pi /15}\int_{0}^{\pi/15}\int_{0}^{cos\theta}\rho^{2} \sin \phi \cos \phi d \rho d \theta d \phi $
A) $\frac { 1 } { 36 } \left( 5 + \cos \frac { 2 \pi } { 15 } \right) \sin ^ { 3 } \frac { \pi } { 15 }$
B) $\frac { 1 } { 6 } \left( 5 + \cos \frac { 2 \pi } { 15 } \right) \sin ^ { 3 } \frac { \pi } { 15 }$
C) $\frac { 1 } { 6 } \left( 3 + \cos \frac { 2 \pi } { 15 } \right) \sin ^ { 2 } \frac { \pi } { 15 }$
D) $\frac { 1 } { 36 } \left( 5 + \cos \frac { \pi } { 15 } \right) \sin ^ { 3 } \frac { \pi } { 15 }$
E) $\frac { 1 } { 36 } \left( 5 + \cos \frac { 2 \pi } { 15 } \right) \sin ^ { 2 } \frac { \pi } { 15 }$

Quizplus · Accepted Answer

The integral can be solved by integrating step by step. First, integrate with respect to $\rho$, treating $\rho$ as the variable and $\theta$ and $\phi$ as constants. The integral of $\rho^2$ from $0$ to $\cos\theta$ is $\frac{1}{3}\cos^3\theta$. Then, integrate the result with respect to $\theta$ from $0$ to $\pi/15$, which does not involve $\phi$, so it's treated as a constant factor. Finally, integrate with respect to $\phi$ from $0$ to $\pi/15$, where the integral of $\sin\phi\cos\phi$ can be simplified using a double angle identity or substitution. The correct answer involves recognizing the integration steps and applying the limits correctly, leading to the formula given in option A.

Evaluate the Following Iterated Integral ∫0π/15∫0π/15∫0cosθρ2sin⁡ϕcos⁡ϕdρdθdϕ \int_{0}^{\pi /15}\int_{0}^{\pi/15}\int_{0}^{cos\theta}\rho^{2} \sin \phi \cos \phi d \rho d \theta d \phi ∫0π/15​∫0π/15​∫0cosθ​ρ2sinϕcosϕdρdθdϕ

Evaluate the Following Iterated Integral $\int_{0}^{\pi /15}\int_{0}^{\pi/15}\int_{0}^{cos\theta}\rho^{2} \sin \phi \cos \phi d \rho d \theta d \phi$