Flux of Vector Field \vec{F} through Surface S

[-EquinoX-]

Homework Statement

Question is:

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

Homework Equations

The Attempt at a Solution

$$\int\limits_R (7x\vec{i} + 7y\vec{j} + (x^2 + y^2)\vec{k}) \cdot (2x\vec{i} + 2y\vec{j} - k) dA$$
$$\int\limits_R (14x^2 + 14y^2 - (x^2 + y^2)) dA$$
$$\int\limits_R (13x^2 + 13y^2) dA$$
$$13 \int\limits_R (x^2 + y^2) dA$$
$$13 \int^{2\pi}_0\int^4_0 r^3 dA$$
$$13 \int^{2\pi}_0 64 dA$$

is this correct?

 

[mighty2000]

Your attempt at solving this problem is correct. You have correctly set up the integral and applied the divergence theorem to calculate the flux of the vector field through the given surface. Your final answer of 832π is also correct. Good job!