Let $C$ be a circle of radius $a$ centered at the origin, traversed counterclockwise. For what nonzero value of $a$ is $\oint_C \bigl(-y + \frac{1}{3} y^3 + x^2 y \bigr) dx = 0$?
We identify $L(x,y) = -y + \frac{1}{3} y^3 + x^2 y$ and $M(x,y) = 0$.
We compute the partial derivatives \begin{align} \partial_y L(x,y) &= -1 + y^2 + x^2\\ \partial_x M(x,y) &= 0 \end{align}
Applying the theorem and computing the partial derivatives, we see that $$\oint_C \bigl(-y + \frac{1}{3} y^3 + x^2 y \bigr) dx = \iint_D (1 - y^2 -x^2) dxdy $$ where D is the disc of radius $a$.