Parameterize C by
x(t) = 2 cos(t) ⇒ dx/dt = -2 sin(t)
y(t) = 2 sin(t) ⇒ dy/dx = 2 cos(t)
with -π/2 ≤ t ≤ π/2. Then the line element is
ds = √((dx/dt)² + (dy/dt)²) dt = 2 dt
and the line integral is
[tex]\displaystyle \int_C xy^4 \, ds = \int_{-\frac\pi2}^{\frac\pi2} 2\cos(t) \cdot (2\sin(t))^4 \cdot (2 \, dt) = 64 \int_{-\frac\pi2}^{\frac\pi2} \cos(t) \sin^4(t) \, dt[/tex]
Substitute u = sin(t) and du = cos(t) dt. Then
[tex]\displaystyle \int_C xy^4 \, ds = 64 \int_{-1}^1 u^4 \, du = \frac{64}5 (1^5 - (-1)^5) = \boxed{\frac{128}5}[/tex]
