Answer:
A = 2x√(36 - x²), -6 <= x <= 6
Explanation:
Since the semicircle is centred at the origin, it will be symmetrical about the y-axis.
So as the top right coordinate of the rectangle is (x, y), then the top left would be (-x, y). We can deduce from this that the length of the rectangle must be 2x.
Similarly, we can deduce that the height of the rectangle is just y.
This means that the area, A = 2xy, but since we want A as a function of x only, we say that A = 2x√(36 - x²).
For the domain, since the function is a square root, assuming this is for real values of x only, then √(36 - x²) => 0
Now we solve for x:
36 - x² => 0
x² <= 36
x = ±6, hence the domain is -6 <= x <= 6
(Obviously you can see this in the diagram but this is how you would actually show it with calculation)