Answer
$x=2 \cos t ; y =2 \sin t$ and $z=2 \sin (2t)$; $0 \leq t \leq 2 \pi$
or,
$x=2 \cos t ; y =2 \sin t$ and $z=4 \sin t \cos t$; $0 \leq t \leq 2 \pi$
Work Step by Step
Given; $x^2+y^2=4$ and radius is $r=\sqrt 4=2$
Write the parametric equations of a circle of radius $2$.
$x=2 \cos t ; y =2 \sin t$
Here, we have $z=xy=(2 \cos t) (2 \sin t)$
or, $z= 4 \sin t \cos t= 2 \sin 2t=2 [2\sin t \cos t]= 2 \sin (2t)$
Our parametric equations are:
$x=2 \cos t ; y =2 \sin t$ and $z=4 \sin t \cos t$; $0 \leq t \leq 2 \pi$
$x=2 \cos t ; y =2 \sin t$ and $z=2 \sin (2t)$; $0 \leq t \leq 2 \pi$
It can also be written as follows:
$x=2 \cos t ; y =2 \sin t$ and $z=4 \sin t \cos t$; $0 \leq t \leq 2 \pi$
