Evaluate the line integral ∫C x(y^4)ds, C is the right half of the circle x2 + y2 = 4 oriented counterclockwise?

Parameterize C by x = 2 cos t, y = 2 sin t with t in [-π/2, π/2].

Then, ∫c xy⁴ ds

= ∫(t = -π/2 to π/2) x(t) y(t)⁴ √((dx/dt)² + (dy/dt)²) dt

= ∫(t = -π/2 to π/2) (2 cos t) (2 sin t)⁴ √((-2 sin t)² + (2 cos t)²) dt

= ∫(t = -π/2 to π/2) (2 cos t) (2 sin t)⁴ * 2 dt

= ∫(t = -π/2 to π/2) 64 sin⁴t cos t dt

= (64/5) sin⁵t {t = -π/2 to π/2}

= 128/5.

I hope this helps!

Ralph Macchio.