Using the Divergence Theorem, find the outward flux of F across the boundary of the region D.
-$\mathbf { F } = 2 \mathrm { x } ^ { 3 } \mathbf { i } + 2 \mathrm { y } ^ { 3 } \mathrm { j } + 2 \mathrm { z } ^ { 3 } \mathbf { k }$; D: the thick sphere $16 \leq \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } \leq 25$
A) $\frac { 50424 } { 5 } \pi$
B) $366 \pi$
C) $50,424 \pi$
D) $984 \pi$

Question

Quizplus · Accepted Answer

The divergence of F is 6x^2 + 6y^2 + 6z^2, so the outward flux of F across the boundary of D is$$\begin{split} \iint_{\partial D} \mathbf{F} \cdot d\mathbf{S} & = \iiint_D 
abla \cdot \mathbf{F} dV \\ & = \iiint_D (6x^2 + 6y^2 + 6z^2) dV \\ & = \int_0^{2\pi} \int_0^{\pi} \int_4^5 (6r^2) (r^2 \sin \phi) dr d\phi d	heta \\ & = \int_0^{2\pi} \int_0^{\pi} 6r^4 \sin \phi d\phi d	heta \\ & = \int_0^{2\pi} \left[ -6r^4 \cos \phi ight]_0^{\pi} d	heta \\ & = \int_0^{2\pi} 12r^4 d	heta \\ & = \left[ \frac{12r^5}{5} ight]_4^5 \\ & = \frac{50424}{5} \pi. \end{split}$$

Using the Divergence Theorem, Find the Outward Flux of F F=2x3i+2y3j+2z3k\mathbf { F } = 2 \mathrm { x } ^ { 3 } \mathbf { i } + 2 \mathrm { y } ^ { 3 } \mathrm { j } + 2 \mathrm { z } ^ { 3 } \mathbf { k }F=2x3i+2y3j+2z3k

Using the Divergence Theorem, Find the Outward Flux of F $\mathbf { F } = 2 \mathrm { x } ^ { 3 } \mathbf { i } + 2 \mathrm { y } ^ { 3 } \mathrm { j } + 2 \mathrm { z } ^ { 3 } \mathbf { k }$