Evaluate the spherical coordinate integral.
-$$\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \pi / 2 } \int _ { 2 } ^ { 4 } \varrho \sin \varphi \mathrm { de } \mathrm { d } \varphi \mathrm { d } \theta$$
A) 3
B) $2 \pi$
C) 2
D) $3 \pi$

Question

Quizplus · Accepted Answer

First, we need to use the conversion formula from spherical coordinates to Cartesian coordinates:
$$x=\rho\sin\varphi\cos\theta,\quad y=\rho\sin\varphi\sin\theta,\quad z=\rho\cos\varphi.$$
We also note that $\rho\sin\varphi$ is the "infinitesimal" element of volume, and we only need to integrate this.
The integral becomes:
$$\int_0^{\pi/2}\int_0^{\pi/2}\int_2^4\rho\sin\varphi\cdot\rho^2\sin\varphi\,d\rho\,d\varphi\,d\theta=\frac{1}{3}\int_0^{\pi/2}\sin^3\varphi\,d\varphi\cdot(4^3-2^3)=\frac{1}{3}\cdot\frac{4}{3}\cdot(4^3-2^3)=3\pi.$$
Therefore, the answer is (D) $3 \pi$.

Evaluate the Spherical Coordinate Integral A) 3 B) 2π2 \pi2π C) 2 D)

Evaluate the Spherical Coordinate Integral A) 3
B) $2 \pi$ C) 2
D)