Evaluate the integral

Question

Evaluate Integral_C xy^(3)dx-4xdy, where C is the portion of the circle centered at the origin of radius 1 in the first quadrant with clockwise rotation.

Answer 1

x = cosθ; y = sinθ; r = 1; and θ from π/2 to 0.

l_C xy^(3)dx-4xdy, = ∫_π/2⁰[cosθsin³θ(-sinθ) - 4cos²θ]dθ = ∫_π/2⁰(- sin⁴θcosθ - 2 - 2cos2θ)dθ =

= (- sin⁵/5 - 2θ - sin2θ)_π/2⁰ = 1/5 +π