## Precalculus (10th Edition)

$\begin{cases} x=\cos t\\ y=4\sin t \end{cases}$ with $-\dfrac{\pi}{2}\leq t\leq \dfrac{\pi}{2}$
The given curve is half an ellipse centered at the origin with the elements: $a=4$ $b=1$ The equation of the ellipse is: $\dfrac{x^2}{1^2}+\dfrac{y^2}{4^2}=1$ $\dfrac{x^2}{1}+\dfrac{y^2}{16}=1$ Find parametric equations that define an ellipse: $\begin{cases} x=b\cos t\\ y=a\sin t \end{cases}$ with $0\leq t\leq 2\pi$ Substitute $a$ and $b$ and determine the domain for the half ellipse: $\begin{cases} x=\cos t\\ y=4\sin t \end{cases}$ The starting point $(0,-4)$: $\begin{cases} \cos t=0\\ 4\sin t=-4 \end{cases}$ $\Rightarrow t=-\dfrac{\pi}{2}$ The ending point $(0,4)$: $\begin{cases} \cos t=0\\ 4\sin t=4 \end{cases}$ $\Rightarrow t=\dfrac{\pi}{2}$ The parametric equations are: $\begin{cases} x=\cos t\\ y=4\sin t \end{cases}$ with $-\dfrac{\pi}{2}\leq t\leq \dfrac{\pi}{2}$