Applying Stoke's Theorem to a parabaloid

[mlb2358]

Homework Statement

Let S be the surface defined by y=10−x^2−z^2 with y≥1, oriented with rightward-pointing normal. Let F=(2xyz+5z)i+e^(x)cosyzj+(x^2)yk. Determine ∫∫∇×F·dS.

Homework Equations

∫∫∇×F·dS = ∫F·dS

The Attempt at a Solution

I think the boundary of the surface is the circle of radius √5 in the xz plane. The parameterization of this should be equal to <√5cost,0,√5sint>. After plugging this parameterization into F and taking the dot product with dS I got ∫-25sin²tdt from 0 to 2 pi, which equals -25∏, however this is not the correct answer. I am not sure about what I am doing wrong. I would appreciate any assistance.

 

[tiny-tim]

hi mlb2358!

(try using the X² button just above the Reply box )

Let S be the surface defined by y=10−x^2−z^2 with y≥1,
…
I think the boundary of the surface is the circle of radius √5 in the xz plane.

nooo …

 

[HallsofIvy]

Reread your problem. What is the restriction on y?

 

[LCKurtz]

And think about the orientation.