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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.

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  • $\begingroup$ Your region is bounded by two curves which you need when you evaluate the double integral. $\endgroup$ – Mhenni Benghorbal Feb 10 '14 at 0:05
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Welcome to MSE, Tony. What you are not grasping is an important hypothesis for Green's theorem to hold: you need a closed curve. What the problem has given you are each component of the curve, and it is a closed one. Therefore, you can apply the Green's theorem and obtain the value.

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.

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