Use a computer algebra system to compute the flux of $\mathbf { F }$ across $S$. $S$ is the surface of the cube cut from the first octant by the planes
$$x = \frac { \pi } { 2 } , y = \frac { \pi } { 2 } , z = \frac { \pi } { 2 }$$
$\mathbf { F } ( x , y , z ) = 5 \sin x \cos ^ { 2 } y \mathbf { i } + 5 \sin ^ { 3 } y \cos ^ { 4 } z \mathbf { j } + 5 \sin ^ { 5 } z \cos ^ { 6 } x \mathbf { k }$
A) $5 \pi ^ { 2 }$
B) $95 \pi ^ { 2 }$
C) $\frac { 5 \pi ^ { 2 } } { 64 }$
D) $\frac { 95 \pi ^ { 2 } } { 64 }$
E) $\frac { 95 \pi ^ { 2 } } { 32 }$

Question

Quizplus · Accepted Answer

Using the Divergence Theorem, we have

$$\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V \operatorname{div} \mathbf{F} dV$$

where $V$ is the region of the cube cut from first octant by the given planes.

We have

$$\operatorname{div} \mathbf{F} = 5(\cos^2 y \cos x + 3\sin^2 y \cos^4 z \cos x + 5\sin^4 z \cos^6 x)$$

Using a computer algebra system (such as WolframAlpha) to perform the integration, we obtain

$$\iiint_V \operatorname{div} \mathbf{F} dV = \frac{95\pi^2}{64}$$

Therefore, the answer is  D.

Use a Computer Algebra System to Compute the Flux Of \mathbf