Multivariable Calculus Green's Theorem Along Two Separate Curves

I have a curve which starts on $(-2,0)$ then goes to $(2,0)$ along the curve $y=4-x^2$ then back to the point $(-2,0)$ along the curve $y=x^2-4$. I have to compute the line integral

$$\int -4x^2y \ dx -(x^3+y^3) \ dy$$

using Green's theorem. I know how to do Green's theorem with the partial derivatives but I don't quite understand how it being along the two curves makes a difference. Would I apply Green's theorem twice? Thank you.

• Your region is bounded by two curves which you need when you evaluate the double integral. – Mhenni Benghorbal Feb 10 '14 at 0:05

To understand how this could be different, consider the case where the exercise gives you only the first curve, i.e. you have the curve $y=4-x^2$ connecting $(-2,0)$ to $(2,0)$. Can you apply Green's theorem? No.