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Let S be the part of the plane z = 2x + y + 5 inside the cylinder x^2 + y^2 = 1 with the normal oriented upwards, and let F := xi - zj - yk.
see attached file for problem. Thanks you

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Solution Summary

This provides examples of finding area, double integrals, and curl.

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Please see the attached file for detailed solution.

a) Let z = f(x,y) be a differentiable surface defined over a region R. Then its surface area is given by

For the surface , its partial derivatives are:

Therefore, the surface area is

The region R in the xy¬-plane is just a disk with radius 1 centered at the origin since that is the region that will lie inside the given cylinder.
So the disk with radius 1 has area of
Then the surface area is

b) Using the divergence theorem,
so first find the divergence of the vector field
then
that is, the above surface ...

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