Question

Use the Divergence Theorem to compute the net outward flux of the vector field

Answer 1

$\vec F(x,y,z)=(x^2,-y^2,z^2)$ has divergence

$\mathrm{div}\vec F(x,y,z)=2x-2y+2z$

so that by the divergence theorem, the flux of $\vec F$ across the boundary of $D$ is

$\displaystyle\iint_{\partial D}\vec F\cdot\mathrm d\vec S=2\iiint_D(x-y+z)\,\mathrm dV$

which you can compute by subtracting [the integral of $\mathrm{div}\vec F$ over the region bounded by $9-x-y$ and the coordinate axes] and [the integral of $\mathrm{div}\vec F$ over the region bounded by $3-x-y$ and the coordinate axes].

$=2\displaystyle\left\{\int_0^9\int_0^{9-x}\int_0^{9-x-y}-\int_0^3\int_0^{3-x}\int_0^{3-x-y}\right\}(x-y+z)\,\mathrm dz\,\mathrm dy\,\mathrm dx$

$=2\left(\dfrac{2187}8-\dfrac{27}8\right)=\boxed{540}$