Apply Green's theorem

Green's theorem in the plane

If C is a simple closed curve (anticlockwise) bounding a region R, and P(x,y), Q(x,y) have continuous first partial derivatives in R, then

∮_C (P dx + Q dy) = ∬_R (∂Q/∂x − ∂P/∂y) dx dy.

Application

P = 2x − y + 4,   Q = 5y + 3x − 6
∂Q/∂x = 3,   ∂P/∂y = −1
∂Q/∂x − ∂P/∂y = 3 + 1 = 4

The triangle has vertices (0,0), (3,0), (3,2). The hypotenuse is y = 2x/3. So x runs from 0 to 3 and y from 0 to 2x/3.

∮_C = ∬_R 4 dx dy = 4 ∫₀³ ∫₀^(2x/3) dy dx
    = 4 ∫₀³ (2x/3) dx
    = 4 · (2/3) · (9/2)
    = 12

Check: area of triangle = ½ · 3 · 2 = 3, and 4 × 3 = 12.

Answer: ∮_C = 12.