Evaluate the line integrals C x sin t y cos t 0 le t le pi2

Evaluate the line integrals C: x= sin t, y = cos t, 0 le t le pi/2 f(x, y) = 2x - y f(x, y)ds =

Solution

Given

x = Sint ; y=Cost

f(x,y) = 2x - y

= 2sint - Cost

Therefore

Integrate f(x,y)dx =  (2sint - Cost)*Costdt

= Integrate(Sint)Costdt - Integrate(Cos²t )

As (d/dt)(sint) = Cost

Therefore

= Sin²t - Integrate(Cos²t )

=  Sin²t - Integrate((1/2)(1 + Cos2t))dt

=  Sin²t - 0.5(t + SintCost) + Constant