# Math Help - Rectangle

1. ## Rectangle

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.

2. I assume that the circle is centred at the origin.

Clearly, its domain is $-6 \leq x \leq 6$ and its range is $0 \leq y \leq 6$.

The coordinate of any point on the semicircle is $\left(x, \sqrt{36 - x^2}\right)$.

Therefore, as you move the point along the semicircle, its length is $x$ in each direction of the horizontal, and it gains a length of $\sqrt{36 - x^2}$ on its vertical.

So the length of the rectangle is $2x$ and the width is $\sqrt{36 - x^2}$.

Therefore its area is

$A = 2x\sqrt{36 - x^2}$ where $0 < x< 6$.

3. Beautifully explained.