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Math Help - Green's Theorem (multiply connected region)

  1. #1
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    Question Green's Theorem (multiply connected region)

    question.

    if R consists of all points on or enclosed by x^2 + y^2 =100 ( oriented counterclockwise) except those points enclosed by (x+2)^2 + y^2 = 1 (oriented clockwise) or (x-2)^2 +y^2 =1 (oriented clockwise).

    Find the integral of bdry(R) of (e^(-x) + 2y)dx +xdy


    Please show me how to solve this problem. Thank you very much.
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  2. #2
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    Quote Originally Posted by kittycat View Post
    question.

    if R consists of all points on or enclosed by x^2 + y^2 =100 ( oriented counterclockwise) except those points enclosed by (x+2)^2 + y^2 = 1 (oriented clockwise) or (x-2)^2 +y^2 =1 (oriented clockwise).

    Find the integral of bdry(R) of (e^(-x) + 2y)dx +xdy


    Please show me how to solve this problem. Thank you very much.
    In this case Green's theorem would say:
    \int_{\partial R} (e^{-x}+2y,x) = \iint_R \partial_1 (x) - \partial_2 (e^{-x}+2y) dA = \iint_A (-1) dA.
    But that quantity is the negative of the area. The area of this region is 100\pi - \pi - \pi = 98\pi. So the line integral must be -98\pi.
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