Can you apply Stoke's theorem to this?

[flyingpig]

Homework Statement

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

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?

 

[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.

 

[flyingpig]

Surface area somehow relates to surface integral...

Never mind, I will trsut in my own answer.

 

[Pi-Bond]

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

 

[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?

 

[flyingpig]

Yeah...?

 

[Pi-Bond]

Never mind, it seems Hurkyl deleted that post.