Let $F$ be the vector field $F=4xz\,i+xz^2\,j+x^7y^3\,k$. Let $S$ be the portion of the paraboloid $z=x^2+y^2$ below $z=9$, along with the disk $x^2+y^2\leq9$ in the plane $z=9$, all with outward orientation. Find $\iint_SF\cdot dS$.
How would I do this? It's one of my HW problems and I'm really struggling.
Portion of the paraboloid $z = x^2 + y^2$ below $z = 9$, along with the disk $x^2 + y^2 \leq 9 \,$ in the plane $z = 9$. That tells you it is a closed surface. So it is easier to go ahead and apply Divergence Theorem (unless you have been asked to do it differently).
$F=4xz\,i+xz^2\,j+x^7y^3\,k$
Divergence of the vector field, Div $F = \nabla \cdot F = \frac{\partial}{\partial x} 4xz + \frac{\partial}{\partial y} xz^2 + \frac{\partial}{\partial z} x^7y^3m = \,?$
Using cylindrical coordinates $x = r \cos \theta, y = r \sin \theta, z = z$
So $dV = r \, dz \, dr \, d \theta$.
As you can see the bound is $r^2 \leq z \leq 9$, $0 \leq r \leq 3, 0 \leq \theta \leq 2 \pi$.
Now set up your integral $\iiint_R (\nabla \cdot F) \, dV$.