Parametrising the surface enclosed by a parametric curve

Question

I have a curve given by $(\cos t, \sin t, \sin 2t)$ with $0 \leq t \leq 2\pi$:

I need to integrate a function over any of the infinitely many surfaces to which this curve is a boundary. How would I find a parametrised form of such a surface?

Answer 1

You can see that $$z(t)=\sin 2t = 2\sin (t) \cos (t)=2x(t)y(t)$$ Therefore, a possible surface is $$ z=2xy, \quad x=x, \quad y=y\quad \mbox{with}\quad x^2+y^2\le 1 $$