Question

A rectangle is bounded by the x-axis and the semicircle y = √36 – x2, as shown in the figure below. Write the area A of the rectangle as a function of x, and determine the domain of the area function.
A =
all real numbers except x = 36
all real numbers except x = 6
0 < x < 6
0 < x < 36
all real numbers

Answer 1

I don't see any "figure below" , but I can surmise your figure shows a quarter circle in the first quadrant with a rectangle, whose base is along the x-axis, its height along the y-axis and it touches the quarter circle at (x,y)

Area = xy
= x√(36-x^2) or x(36-x^2)^(1/2)

for the domain, we have to make sure that the number under the √ does not become negative, so
0 < x < 6