Question

C xy2 ds, c is the right half of the circle x2 y2 = 4 oriented counterclockwise

Answer 1

Parameterize the contour by

$C:\mathbf r(t)=(2\cos t,2\sin t)\implies \dfrac{\mathrm d\mathbf r}{\mathrm dt}=(-2\sin t,2\cos t)$
$\implies\mathrm dS=\sqrt{\left(\dfrac{\mathrm dx}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm dy}{\mathrm dt}\right)^2}\,\mathrm dt=2$

where $-\dfrac\pi2\le t\le\dfrac\pi2$. Then the line integral is given by

$\displaystyle\int_C xy^2\,\mathrm dS=2\int_{-\pi/2}^{\pi/2}(2\cos t)(2\sin t)^2\,\mathrm dt$
$=\displaystyle16\int_{-\pi/2}^{\pi/2}\cos t\sin^2t\,\mathrm dt=\dfrac{32}3$