Use stokes' theorem to find I = [tex]\int\int (\nabla x F) n dS[/tex] where D is the part of the sphere [tex]x^2 + y^2 + (z-2)^2 = 8[/tex] that lies above the xy plane, and
[tex]F=ycos(3xz^2)i + x^3e^[-yz]j - e^[zsinxy]k[/tex]
Attempt at solution:
I want to use the line integral [tex]\int F dr[/tex] to solve this.
I parametrize the boundary [tex]r(t) = (2cos\theta)i + (2sin\theta)j + 0k)[/tex] (the sphere on the xy plane)
Then F dotted with r'[t] is [tex]\int -4(sin\theta)^2 + (2cos\theta)^4 [/tex]
and 0 < theta < 2pi
And the answer that I get is 8pi. Is this correct? More specifically is my boundary curve legit, considering that the sphere "bulges" out above this boundary? :S