Boundary of part of an elliptical paraboloid

Question

Given a certain function $F$, and that $S$ is part of the elliptical paraboloid $$z = x^2 + 4y^2 ~~,~~z \leq 1$$ How do I show that the boundary of $S = C : r(t) = (cost)i~+~(\frac{1}{2}sint)j~+~k$?

Answer 1

Your boundary is given by the parametric equations \begin{cases} x(t)=\cos t\\ y(t)=\frac{\sin t}{2}\quad \quad t \in [0,2\pi]\\ z(t) = 1 \end{cases}

It is easy to see that these equations verify the equation of the paraboloid for all $t\in [0,2\pi]$: $$ x^2(t)+4y^2(t) = 1 = z(t) $$