Question

Calculate the following line integral:
∫ C xy^3dx, where C is the unit circle x^2+y^2=1, oriented counterclockwise. Note: C is at the bottom right of the integral.

Answer 1

Counterclockwise parametrization of $C$ is $(\cos t,\sin t)$ , $t\in[0,2\pi]$ , so

$\int\limits_C xy^3dx=\int\limits_0^{2\pi} \cos t\sin^3 td(\cos t)=$

$=-\int\limits_0^{2\pi} \cos t\sin^4 tdt=-\int\limits_0^{2\pi}\sin^4 td(\sin t)=-\frac{1}{5}\sin^5 t\bigl|_0^{2\pi}=$

$=-\frac{1}{5}\sin^5 2\pi+\frac{1}{5}\sin^5 0=0$

Answer: