My question is:

Compute the flux of the vector field, $\displaystyle \vec{F}$ , through the surface, S.

$\displaystyle \vec{F} = 7\vec{r}$ and S is the part of the surface $\displaystyle z = x^2 + y^2$ above the disk $\displaystyle x^2 + y^2 \leq 4$ oriented downward.

My answer so far is

$\displaystyle \int\limits_R (7x\vec{i} + 7y\vec{j} + (x^2 + y^2)\vec{k}) \cdot (2x\vec{i} + 2y\vec{j} - k) dA $

$\displaystyle \int\limits_R (14x^2 + 14y^2 - (x^2 + y^2)) dA$

$\displaystyle \int\limits_R (13x^2 + 13y^2) dA$

$\displaystyle 13 \int\limits_R (x^2 + y^2) dA$

$\displaystyle 13 \int^{2\pi}_0\int^2_0 r^3 dr d\theta$

$\displaystyle 13 \int^{2\pi}_0 4 d\theta$

$\displaystyle 104\pi$

is this correct?