# Green's Theorem (multiply connected region)

• Nov 13th 2007, 05:39 AM
kittycat
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.
• Nov 13th 2007, 06:47 AM
ThePerfectHacker
Quote:

Originally Posted by kittycat
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:
$\displaystyle \int_{\partial R} (e^{-x}+2y,x) = \iint_R \partial_1 (x) - \partial_2 (e^{-x}+2y) dA = \iint_A (-1) dA$.
But that quantity is the negative of the area. The area of this region is $\displaystyle 100\pi - \pi - \pi = 98\pi$. So the line integral must be $\displaystyle -98\pi$.