F = (2x^2 + 3y) i + (2x + 3y^2) j , around triangle with points (2,0) (0,3) (-2,0)
what are the limits of integration to be used?
Green's theorem says that the integral around a closed curve of Ldx+ Mdy is the integral over the region enclosed by the curve of $\displaystyle \frac{\partial M}{\partial x}- \frac{\partial L}{\partial y}$. Here, the region is the triangle with vertices (2, 0), (0, 3), (-2, 0), a triangle with the x-axis as base, one side from (-2, 0) to (0, 3), which has equation y= (3/2)x+ 3, and another from (0, 3) to (2, 0), which has equation y= -(3/2)x+ 3.
If you want to do the integral in the order "dy dx", then you will need to do it as two integrals. integrate with x going from -2 to 0, y going from 0 to the (3/2)x+ 3, then with x going from 0 to 3, y going from 0 to -(3/2)+ 3.
It would be simpler to do it in the order "dx dy". Then we would write the two equations as x= (2/3)y- 2 and x= -(2/3)y+ 2. y can go from its lowest value, 0, to its highest value, 3. And, for each y, x goes from the left side, (2/3)y- 2, to the right side, -(2/3)y+ 2.