Use Green's Theorem to evaluate the line integral along the given positively oriented curve.

Question

integral of xy2 dx + 4x2y dy

C is the triangle with vertices (0, 0), (2, 2), and (2, 4)

Answer 1

Green's Theorem says that

∮_CMdx+Ndy = ∫∫_RMx+Ny dx dy

Nx = 8xy
My = 2xy
So, we end up with
∫[0,2]∫[x,2x] 10xy dy dx
= ∫[0,2] 5xy^2 [x,2x] dx
= ∫[0,2] 5x(4x^2-x^2) dx
= ∫[0,2] 15x^3 dx
= 5x^4 [0,2]
= 80

Answer 2

Oops. That is Nx - My

I'll let you make the change and redo the arithmetic.