1. ## Green's Theorem

Use Green's Theorem to evaluate the line integral along the given positively oriented curve:

$\displaystyle \int_{C}F \cdot dr, F = (y^2 - x^{2}y)i + xy^2j$

C consists of the circle $\displaystyle x^2 + y^2 = 16$ from (4, 0) to $\displaystyle (2\sqrt{2}, 2\sqrt{2})$ and the line segments from $\displaystyle (2\sqrt{2}, 2\sqrt{2})$ to (0, 0) and from (0, 0) to (4, 0).

2. Originally Posted by wik_chick88
Use Green's Theorem to evaluate the line integral along the given positively oriented curve:

$\displaystyle \int_{C}F \cdot dr, F = (y^2 - x^{2}y)i + xy^2j$

C consists of the circle $\displaystyle x^2 + y^2 = 16$ from (4, 0) to $\displaystyle (2\sqrt{2}, 2\sqrt{2})$ and the line segments from $\displaystyle (2\sqrt{2}, 2\sqrt{2})$ to (0, 0) and from (0, 0) to (4, 0).
$\displaystyle \int_{C}F \cdot dr =\int\int_{D}( \frac {\partial {(xy^2)}}{\partial x}-\frac {\partial {(y^2-x^{2}y)}}{\partial y}) dxdy$, where D is the region enclosed by the curve, C. Then calculate the double integral in polar coordinates.