evaluate the integral where C is the...

Question

I just want to check the answer.Thank you!

Evalulate $$\int_C (y+\sin x)dx + (z^2+\cos y)dy +x^3dz$$ where $C$ is the curve given parametrically by $$r(t) = \langle \sin t, \cos t, \sin 2t \rangle$$, $0\leq t\leq 2\pi$

Answer 1

Note that for any closed curve $\gamma:\>t\mapsto {\bf r}(t)$ $(0\leq t\leq 2\pi)$ one has $$\int_\gamma \sin x \>dx=\int_0^{2\pi}\sin\bigl(x(t)\bigr)\,\dot x(t)\>dt=-\cos\bigl(x(2\pi)\bigr)+\cos\bigl(x(0\bigr)=0\ ,$$ and similarly $\int_\gamma \cos y\>dy=0$. It remains to evaluate $$J:=\int_C\bigl(y\,dx+z^2\,dy+x^3\,dz)\ .$$ For the given $C:\>t\mapsto{\bf r}(t):=\bigl(\sin t,\cos t, \sin(2t)\bigr)$ we have $\dot{\bf r}(t)=\bigl(\cos t, -\sin t, 2\cos(2t)\bigr)$. It follows that $$J=\int_{-\pi}^\pi\bigl(\cos^2 t+\sin^2(2 t)(-\sin t)+2\sin^3 t\>\cos(2t)\bigr)\>dt=\pi\ ,$$ since only the first term under the integral sign gives a contribution, by symmetry.