A Helix

The arc length $S$ is given by $S = \int \sqrt{(dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2} dt$ , or $\frac{dS}{dt} = \sqrt{(dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2} \Rightarrow dS = \sqrt{(dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2} dt.$ Turning now to our path integral, we can now write:

$\int_{C} xy^3 dS = \int_{0}^{\pi/2} (4\sin t)(4 \cos t)^3 \cdot \sqrt{(4\cos t)^2 + (-4\sin t)^2 + 3^2} dt$ ;

or $\int_{0}^{\pi/2} 256\sin t \cos^{3} t \cdot \sqrt{16 + 9} dt$ ;

or $1280 \int_{0}^{\pi/2} \sin t \cos^{3} t dt$ ;

or $-320\cos^{4}t|_{0}^{\pi/2} = \boxed{320}.$