I have attempted to calculate the line integral
$$\int\limits_L \left(3y+e^{\cos(x)}\right)\ dx + \left(7x-\sqrt[3]{y^4+10}\right)\mathrm{d}y$$
where $L$ is the circle $x^2+y^2=3^2$.
I decided to use Green's theorem by calculating the partial derivatives and subtracting.
$$\frac{\partial}{\partial x}\left(7x-\sqrt[3]{y^4+10}\right) = 7$$
$$\frac{\partial}{\partial y}\left(3y+e^{cos(x)}\right) = 3$$
$$\int\limits_L \left(3y+e^{cos(x)}\right)\ dx + \left(7x-\sqrt[3]{y^4+10}\right)\ dy = \iint\limits_D 4\ dx\ dy$$
It seemed to me at this point, that I should use polar coordinates since we are dealing with a circle.
I let $\theta$ be bounded by $0$ and $2\pi$ and $r$ be from $0$ to $3$ (the radius of the circle).
When doing change of variables, I let the Jacobian be $r$ to get:
$$\int_0^3 \int_0^{2\pi} 4r\ d\theta\ dr$$
Which comes out to $36\pi$ in the end.
Have I done the change of variables correctly?
Is there a way to check that I got the right answer?