I will be using the S sign to portray the integral sign.
1) S_c F*dr where F(x,y) = (y^2 - x^2 y)i + xy^2 j and C consists of the circle x^2 + y^2 = 4 from (2, 0) to (sqrt(2), sqrt(2)) and line segment from ( sqrt(2), sqrt(2) ) to (0, 0) and from (0, 0) to (2, 0).
2) S_c F*dr where F(x, y) = x^3 yi + x^4 j and C is the cruve x^4 + y^4 = 1.
Let L = y^2 - x^2.y and M = xy^2; then M_x - L_y = y^2 - (2y - x^2) = x^2 + y^2 - 2y,
where subscripts denote partial derivatives. Now, according to Green's theorem, the integral is
int (x^2 + y^2 - 2y) dx dy
over the ...
This provides two examples of using Green's theorem to evaluate a line integral along a curve.