Use divergence theorem to find the flux of F across surface

H

hank

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Sep 13, 2006
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..with outward orientation.

F = x^3 i + yx^2 j + xy k; the surface of the solid bounded by z = 4 - x^2, y + z = 5, z = 0, y = 0.

I get div F = 3x^2 + x^2 = 4x^2, and

= SSS 4x^2 dzdydz

I'm completely lost how to find the limits of integration on this.
My best guess was -2 <= x <= 2, 0 <= y <= 5, and 0 <= z <= 4 - x^2, which naturally is wrong.

The sketch I made of the surface doesn't make any sense to me either.

Thanks for any help you can give.
 
Someone correct me if I'm wrong, but I think your limits look pretty good except the lower bound of y.
x is bounded by the intersection of z = 4 - x^2 and the xy-plane which occurs at z=0. Solving 0 = 4 - x^2 for x gives +-2.
y is bounded by the intersection of the plane y + z = 5 and the xy-plane, and the intersection of y + z = 5 and the top of z = 4 - x^2. The former occurs at z = 0 which gives y + 0 = 5, and the latter occurs at z = 4 - 0^2 which gives y + 4 = 5 or y = 1.
And z is obviously bounded by z = 0 and z = 4 - x^2.
 
Thanks for the reply.

I came up with that limit at one point myself, where 0 <= y <=1 , but my answer after integrating was way off. My answer came out to 136/15 and the book says the answer is 4608/35.
 
Hm, I think you misunderstood me. 1 <= y <= 5 was my suggestion.
Edit: Wait, y=0 is the given constraint in the problem, sorry. And z should depend on y as well, not just x. :? I give up.
 
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