The vectorfield F is given by $$\mathbf{F}(x,y) = (x^3 - y, x + y^3) $$ Calculate $$ \oint_C \mathbf{F} \cdot d\mathbf{r}$$ Where C is the boundary of the region enclosed by $y = x$ and $y = x^2$ and C is oriented counter clockwise.
I cant seem to find the right bounds for $r$. I thought it would make sense to let $r$ range from $0$ to $\frac{1}{cos\theta}$ but this gives an incorrect answer. $\theta$ will of course range from $0$ to $2\pi$ and the integrand is $2r$.
The region bounded by $y=x$ and $y=x^2$ is described as $ 0\le \theta \le \pi /4$, and $ 0\le r\le \frac {\sin \theta }{ \cos ^2 \theta }$
Note that $y=x^2$ in polar form is $ r\sin \theta = r^2 \cos ^2 \theta.$