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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 Preview

    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
    that is, the above surface ...

    Solution Summary

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