Surface Integral using Divergence Theorem

[Saladsamurai]

Homework Statement

Evaluate the surface Integral $$I=\int\int_S\vec{F}\cdot\vec{n}\,dS$$

where $$\vec{F}=<z^2+xy^2,x^100e^x, y+x^2z>$$

and S is the surface bounded by the paraboloid $z=x^2+y^2$

and the plane z=1; oriented by the outward normal.

The Attempt at a Solution

$$I=int\int_S\vec{F}\cdot\vec{n}\,dS=\int\int\int_E(div\vec{F})dV$$

$$(div\vec{F})=y^2+x^2$$

$$\Rightarrow I=\int\int_D(\int_{z=x^2+y^2}^1(x^2+y^2)\,dz)\,dA$$

$$\Rightarrow I=\int\int_D(1-(x^2+y^2)\,dA$$

Is it just Polar Coordinates all the way home now?

Thanks,
Casey

 

[Dick]

No, I don't think that's right. The integral of f(x,y)dz is z*f(x,y), isn't it? Try evaluating that between the limits again.

 

[Saladsamurai]

I am sorry, where are you referring to?
The divergence of F= x^2+y^2 That is f(x,y), now if I integrate that wrt z
ohh! I forgot to include the factor of f(x,y) so it should be

$$I=\int\int_D(x^2+y^2)*(1-(x^2+y^2)\,dA$$

yes? Now Polar?

 

[Dick]

Now polar.

 

[Saladsamurai]

Polar bears. Polar beers. Sweet numchuck skills Dick. Thanks.