- #1

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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 x-axis 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.