Answer:

![x \in [-6,6]](https://img.qammunity.org/2024/formulas/mathematics/high-school/ol1opkdjbk5jp6t4lpv4faiwfyph8cm2yq.png)
![y \in [0,3]](https://img.qammunity.org/2024/formulas/mathematics/high-school/63fszjsigmitwfxx4mjxhcsauksnrjt18j.png)
Explanation:
Given parametric equations:


To eliminate the parameter θ, rewrite x in terms of cosθ, then substitute this into the second equation to find a Cartesian equation for the curve.
Rearrange the first equation to isolate sinθ:

Substitute this into the trigonometric identity sin²θ + cos²θ = 1, then rearrange to isolate cosθ:

Substitute the expression for cosθ into the equation for y:

Since 0 ≤ θ ≤ π, we know that θ is in the first and second quadrants (above the x-axis). Therefore, the Cartesian equation for the curve is:

The expression under the square root sign should always be greater than or equal to zero. Therefore:


Therefore:

So the domain (values of x) is limited to [-6, 6].
When x = ±6, y = 0. Therefore, the minimum value of y is zero.
The maximum value of y will be the value when the value under the square root sign is at its maximum, so when x = 0:

So the range (values of y) is limited to [0, 3].
Therefore:
![\large\boxed{\textsf{$y=(1)/(2)√(36-x^2)$\;\;where\;\;$x \in [-6,6]$\;\;and\;\;$y \in [0,3]$}}](https://img.qammunity.org/2024/formulas/mathematics/high-school/sz2pcpsrgfhtj2bc466f5z6tiovoy2z9n3.png)