# Can you apply Stoke's theorem to this?

1. Jul 14, 2011

### flyingpig

1. The problem statement, all variables and given/known data

What is the surface area of a hemi-sphere of radius 6 centered at the origin above the xy-plane lying outside the cylinder r2 = 9

3. The attempt at a solution

I did a lot of algebra using the hard way by setting $$z = \sqrt{6^2 - x^2 - y^2}$$ and use the formula

$$A(S) = \iint \sqrt{1 + \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2} dA$$

Did a lot of messy algebra and got 324π

Is there an easier way to do this?

2. Jul 14, 2011

### Pi-Bond

If you could, wouldn't that imply that the surface area of any figure with the same bounding curve be 324π? Also I don't see how you would get the surface area in terms of curl of a vector field.

3. Jul 14, 2011

### flyingpig

Surface area somehow relates to surface integral...

Never mind, I will trsut in my own answer.

4. Jul 14, 2011

### Pi-Bond

Yes, but not necessarily to the curl of a vector field.

5. Jul 14, 2011

### Pi-Bond

Stoke's theorem says that the surface integral of the curl of a vector field is equal to the loop integral of the vector field over the bounding curve, right?

6. Jul 14, 2011

### flyingpig

Yeah....?

7. Jul 14, 2011

### Pi-Bond

Never mind, it seems Hurkyl deleted that post.