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
$$\begin{array} { l }
x = \frac { \pi } { 2 } , y = \frac { \pi } { 2 } , z = \frac { \pi } { 2 } \
\mathbf { F } ( x , y , z ) = 3 \sin x \cos ^ { 2 } y \mathbf { i } + 3 \sin ^ { 3 } y \cos ^ { 4 } z \mathbf { j } + 3 \sin ^ { 5 } z \cos ^ { 6 } x \mathbf { k }
\end{array}$$
Select the correct answer.
A) $\frac { 3 \pi ^ { 2 } } { 64 }$
B) $\frac { 57 \pi ^ { 2 } } { 64 }$
C) $57 \pi ^ { 2 }$
D) $\frac { 57 \pi ^ { 2 } } { 32 }$
E) $3 \pi ^ { 2 }$

Question

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
$$\begin{array} { l } 
x = \frac { \pi } { 2 } , y = \frac { \pi } { 2 } , z = \frac { \pi } { 2 } \
\mathbf { F } ( x , y , z ) = 3 \sin x \cos ^ { 2 } y \mathbf { i } + 3 \sin ^ { 3 } y \cos ^ { 4 } z \mathbf { j } + 3 \sin ^ { 5 } z \cos ^ { 6 } x \mathbf { k }
\end{array}$$
Select the correct answer.
A) $\frac { 3 \pi ^ { 2 } } { 64 }$
B) $\frac { 57 \pi ^ { 2 } } { 64 }$
C) $57 \pi ^ { 2 }$
D) $\frac { 57 \pi ^ { 2 } } { 32 }$
E) $3 \pi ^ { 2 }$

Quizplus · Accepted Answer

The flux is computed using the divergence theorem. The divergence of $\mathbf{F}$ is $3 \cos^2 y \cos^6 x + 3 \sin^2 y \cos^4 z + 3 \sin^4 z \cos^6 x$. Integrating this over the volume of the cube $[0, \frac{\pi}{2}]^3$ gives $\frac{57 \pi^2}{64}$.

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