A rectangle is bounded by the x-axis and the semicircle
y = sqrt[36 - x^2] . Write the area A of the rectangle as a function of x, and determine the domain of the function.
I assume that the circle is centred at the origin.
Clearly, its domain is $\displaystyle -6 \leq x \leq 6$ and its range is $\displaystyle 0 \leq y \leq 6$.
The coordinate of any point on the semicircle is $\displaystyle \left(x, \sqrt{36 - x^2}\right)$.
Therefore, as you move the point along the semicircle, its length is $\displaystyle x$ in each direction of the horizontal, and it gains a length of $\displaystyle \sqrt{36 - x^2}$ on its vertical.
So the length of the rectangle is $\displaystyle 2x$ and the width is $\displaystyle \sqrt{36 - x^2}$.
Therefore its area is
$\displaystyle A = 2x\sqrt{36 - x^2}$ where $\displaystyle 0 < x< 6$.