Q$4.(4$ points$)$ Consider the curve $x$:$[0,2]R^2,$ defined by

$x(t)=(cos(t),\frac{1}{2}sin(2t))$

a$)(1$ point$)$ Is the curve closed? Is the curve simple?

b$)(1$ point$)$ Show that for any $ain{R}^2,{\displaystyle \underset{0}{\overset{2}{}}}a*x^{\text{'}}(t)dt=0.$

c$)(2$ points$)$ More generally, prove that for any closed $C^1$ curve $y$:$[a,b]R^n$ and any vector $ain{R}^n,$

${\displaystyle \underset{a}{\overset{b}{}}}a*y^{\text{'}}(t)dt=0$