Examples

Triple integral can be easier than six flux integrals

Triple integral can be easier than six flux integrals

Problem: A box $\mathcal{W}$ is given by $(x,y,z)\in [0,2]\times [0,3]\times [0,1]$. Calculate the total flux leaving this box for the vector field $\mathbf{F}=(x^2,z^4,e^z)$.

Solution: Instead of computing the vector surface integral for all six sides of the box, we compute the triple integral over the interior of the box. This equals the desired flux by the divergence theorem.

First find the divergence:

$$\nabla \cdot \mathbf{F}=\frac{\partial }{\partial x}{x}^2+\frac{\partial }{\partial y}{z}^4+\frac{\partial }{\partial z}{e}^z=2x+e^z.$$

Now integrate for the answer:

$$\begin{array}{rl}{\iiint }_{\mathcal{V}}\nabla \cdot \mathbf{F}dV& ={\int }_0^1{\int }_0^2{\int }_0^3(2x+e^z)dydxdz\\ & =3{\int }_0^1{\int }_0^2(2x+e^z)dxdz\\ & =3\cdot 2^2+3\cdot 2\cdot (e^1-1)=6(e+1).\end{array}$$