Evaluate the spherical coordinate integral.
-$\int_{0}^{\pi / 2} \int_{0}^{\pi / 2} \int_{0}^{9 \cos \varphi} \varrho^{2} \sin \varphi \mathrm{de} d \varphi d \theta$
A) $\frac { 81 } { 2 } \pi ^ { 2 }$
B) $\frac { 243 } { 8 } \pi$
C) $\frac { 81 } { 2 } \pi$
D) $\frac { 243 } { 8 } \pi ^ { 2 }$

Question

Evaluate the spherical coordinate integral.
-$\int_{0}^{\pi / 2} \int_{0}^{\pi / 2} \int_{0}^{9 \cos \varphi} \varrho^{2} \sin \varphi \mathrm{de} d \varphi d \theta$ 
A) $\frac { 81 } { 2 } \pi ^ { 2 }$
B) $\frac { 243 } { 8 } \pi$
C) $\frac { 81 } { 2 } \pi$
D) $\frac { 243 } { 8 } \pi ^ { 2 }$

Quizplus · Accepted Answer

The integral evaluates to $\frac{243}{8} \pi$ by integrating $\varrho^2 \sin \varphi$ over the given limits.

Evaluate the Spherical Coordinate Integral ∫0π/2∫0π/2∫09cos⁡φϱ2sin⁡φdedφdθ\int_{0}^{\pi / 2} \int_{0}^{\pi / 2} \int_{0}^{9 \cos \varphi} \varrho^{2} \sin \varphi \mathrm{de} d \varphi d \theta∫0π/2​∫0π/2​∫09cosφ​ϱ2sinφdedφdθ

Evaluate the Spherical Coordinate Integral $\int_{0}^{\pi / 2} \int_{0}^{\pi / 2} \int_{0}^{9 \cos \varphi} \varrho^{2} \sin \varphi \mathrm{de} d \varphi d \theta$