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 and its range is .
The coordinate of any point on the semicircle is .
Therefore, as you move the point along the semicircle, its length is in each direction of the horizontal, and it gains a length of on its vertical.
So the length of the rectangle is and the width is .
Therefore its area is
where .