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Homework Statement
Let F = <x, z, xz> evaluate ∫∫F⋅dS for the following region:
x^{2}+y^{2}≤z≤1 and x≥0
Homework Equations
Gauss Theorem
∫∫∫(∇⋅F)dV = ∫∫F⋅dS
The Attempt at a Solution
This is the graph of the entire function:
Thank you Wolfram Alpha.
But my surface is just the half of this paraboloid where x is positive. So I thought if I looked down the xaxis I would get something like this:
But only the right half of the circle (from 3π/2 to π/2)...
The integral I set up is the following:
∫∫∫xdzdxdy (x is the dot product of ∇ and F)
I converted to polar coordinates
∫∫∫r^{2}cosθdzdrdθ
Bounds:
r^{2}≤z≤1
0≤r≤1
3π/2≤θ≤π/2
I ended up getting
(4/15)∫cosθdθ
3π/2≤θ≤π/2
(4/15)[sin(π/2)sin(3π/2)] = 0
Answer should be 4/15 according to the back of the book.
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