Evaluate 12 integralC ydx x dy directly as a line integral

Evaluate 1/2 integral_C - ydx + x dy directly as a line integral, where C is the ellipse x^2/4 + y^2/9 = 1 parametrized by x = 2 cos (t), y = 3 (t), 0 lessthanorequalto t lessthanorequalto 2 pi. Apply GREEN\'S THEOREM to find the area enclosed by the curve C of part (a).

Solution

x = 2*cost =======> dx = -2*sint*dt

y = 3*sin t =======> dy = 3*cost*dt

I = 0.5*[6*sin^2 t + 6*cos^2 t]dt [0 to 2pi]

I = 0.5*6*dt = 3*t = 3*[2pi - 0] = 6pi