How to Use Stokes' Theorem for Evaluating Line Integrals?

[jegues]

Homework Statement

Use Stoke's theorem to evaluate the line integral

$$\oint y^{3}zdx - x^{3}zdy + 4dz$$

where C is the curve of intersection of the paraboloid $$z = 2 + x^{2} + y^{2}$$ and the plane $$z=5$$, directed clockwise as viewed from the point (0,0,7).

Homework Equations

The Attempt at a Solution

Am I doing everything up to this integral correctly? I'm stuck at where I'm at now.

Did I make a mistake along the way?

EDIT: I found one mistake, since my surface S is Z=5, my dS should simply be (1)dA. I fixed my mistakes and found the answer to be $$\frac{135\pi}{2}$$

 

[LCKurtz]

Homework Statement

Use Stoke's theorem to evaluate the line integral

$$\oint y^{3}zdx - x^{3}zdy + 4dz$$

where C is the curve of intersection of the paraboloid $$z = 2 + x^{2} + y^{2}$$ and the plane $$z=5$$, directed clockwise as viewed from the point (0,0,7).

Homework Equations

The Attempt at a Solution

Am I doing everything up to this integral correctly? I'm stuck at where I'm at now.

Did I make a mistake along the way?

EDIT: I found one mistake, since my surface S is Z=5, my dS should simply be (1)dA. I fixed my mistakes and found the answer to be $$\frac{135\pi}{2}$$

Check your z component of curl F. You should be able to factor out a z. And with n=-k the integral should come out very easy.