Green's theorem with vector field

Question

I am given the vector field $$\vec{F} = (9x^2y+3y^3+2e^x,6e^{y^2}+225x)$$ and $C_a$ a circle of radius a and the center of origin (0,0), counter clockwise. I am trying to calculate $$\oint_{C_1} \vec{F}d\vec{r}$$ for a=1, and then I have to find a such that the work would be maximal. What I tried to do is calculate $P_y = 9x^2 + 9y^2$, $Q_x = 225$ so I think the integral should be (after parametrization and using Jacobian: $$\oint_{C_1} \vec{F}d\vec{r} = \int_{0}^{2\pi}\int_0^1(225-9)rdrd\theta = 216\pi$$ and for the second part because it is clockwise the work shoult be maximal the smaller $a$ is but I am not sure how to calculate it. What am I doing wrong and how to find such $a$?

Answer 1

So I forgot that $$x = rcos(\theta)$$ $$y = rsin(\theta)$$ so we get $$\oint_{C_1} \vec{F}d\vec{r} = \int_{0}^{2\pi}\int_0^1(225r - 9r^3)drd\theta = \pi(225- \dfrac 9 2)$$ As for the maximal $a$ we will do $$(225a - 9a^3 = 0)$$ so a maximal is:$$a = \dfrac {\sqrt{225}} 3$$