$displaystyleint_C (e^x+cos(x)+2y),dx+(2x-frac{y^2}{3}),dy$ in an ellipse

Question

I have to compute $displaystyleint_C (e^x+cos(x)+2y),dx+(2x-frac{y^2}{3}),dy$ 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.
$alpha(t)=(2+7cos(t),3+2sin(t)),; tin[0,2pi]$.
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?

Matthew Pilling · Accepted Answer

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, $$int_{C}vec{nabla} f cdot dvec{r}=int int _{R} 0cdot dA=0$$ Here $C$ is your ellipse and $R$ is its interior.

Correct answer by Matthew Pilling on January 17, 2021

$displaystyleint_C (e^x+cos(x)+2y),dx+(2x-frac{y^2}{3}),dy$ in an ellipse

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