Use Green's Theorem to find the integral of a triangle

Question

Use Green's Theorem to evaluate the following integral:

Where C is the triangle with vertices (0, 0), (4, 0), and (2, 3) (in the positive direction).

I have drawn the triangle but I'm having trouble finding the limits for y. Any help would be great!

Answer 1

Please read the Green's Theorem again. If your vector filed is defined in a closed curve, then your line integral can be converted to the double integral. In this question the curve C is closed since in this question C is a triangle, it 's closed. In your line integral which is the dot product of your vector field (F=<P(x,y),Q(x,y)> )and ds (<dx,dy>). As you see form the Green's theorem, this will be equal to the double integral over inside the triangle and Qx-Py= 9 dA (A is the closed region) (dA=dxdy or dydx). Now try to evaluate this double integral. I hope you can figure out the boundaries of x and y then apply either type1 or type 2. (Hint: If you know how the evaluate the area of this triangle then question will be a piece of cake)