Question

Use Greens theorem to calculate the line integral H C x 2ydx + (y − 3)dy,, where C is a rectangle with vertices (1, 1), (4, 1), (4, 5) and (1, 5), oriented counterclockwise.

Answer 1

$Given, \oint_C[\{M(x)+2y\}dx- x^2dy ]\\ Let \space m= M(x) +2y \space and \space N=-x^2\\ \therefore \frac{\partial }{\partial y}(m)=2 \space and \space \frac{\partial }{\partial x}(N)=-2x$

Using the Green theorem

$\therefore \oint_C[\{M(x)+2y\}dx- x^2dy ]=\iint_D[-2x-2]dA\\ =\intop_1^5\int_1^4(-2x-2)dxdy\\ =\int _1^5\int _1^4\left(-2x-2\right)dxdy\\ =\int _1^5\left(-21\right)dy\\ =-84$