Question

calc 3 help

use the divergence theorem to find the flux of F(x, y, z) = (x^3-e^y)i+(y^3+sinz)j + (z^3 - xy)k across the surface theta with outward orientation where theta is the surface of the solid bounded above by z = sqrt(4-x^2-y^2) and below by the xy-plane.

Answer 1

If you could included steps, that would be greatly appreciated.

Answer 2

According to the divergence teorem, the flux is equal to the triple integral:\[\int\limits_{}^{}\int\limits_{}^{}\int\limits_{\theta}^{}Div \vec{F}dV\]First compute the divergence of F. \[Div \vec{F}=3x^2+3y^2+3z^2\]Now, set up the bounds for the region theta and compute the integral.

Answer 3

The region theta intersects the xy-plane in a disc \[x^2+y^2 \le 4\] The z-coordinate goes from zero to \[z=\sqrt {4-x^2-y^2}=\sqrt{4-(x^2+y^2)}\]