I have to compute $\displaystyle\int_C (e^x+\cos(x)+2y)\,dx+(2x-\frac{y^2}{3})\,dy$ in the ellipsoide $\frac{(x-2)^2}{49}+\frac{(y-3)^2}{4}=1$ using Green's Theorem.
The first thing I did was getting the parametric equation of the ellipse.
$\alpha(t)=(2+7\cos(t),3+2\sin(t)),\; t\in[0,2\pi]$.
Now, I'm not sure if I have to replace in the function every $x$ and $y$ with the result I got, and then using (maybe) polar coordinates to finally compute the result. Is my approach correct?
Notice the gradient of $$f(x,y)=e^x+\sin(x)+2xy-\frac{y^3}{9}$$ is the vector field $$\bigg<e^x+\cos(x)+2y,2x- \frac{y^3}{3} \bigg>$$ This means your vector field is conservative, so its integral over any closed curve (like your ellipse, not ellipsoid) is zero. With Green's Theorem, $$\int_{C}\vec{\nabla} f \cdot d\vec{r}=\int \int _{R} 0\cdot dA=0$$ Here $C$ is your ellipse and $R$ is its interior.