SOLUTION: A rectangle is bounded by the x-axis and the semicircle y=(4-x2)square root. Write the area A of the rectangle as a function of x and determine the domain of the function.

 

As you can see by the picture the rectangle has length 2x
(the horizontal measurement) and width y (the vertical
measurement).

              Area = length·width

              Area = (2x)·y

But since y = ,

              Area = (2x)·

              Area =  2x 

And now since we have area expressed as a function of x, we can write

              A(x) =  2x

The domain of 2x is the set of all values of x for
which 2x is a real number, which will be whenever

              4 - x² > 0

      (2 - x)(2 + x) > 0

Critical numbers are 2 and -2, so we have to get test points in
the intervals (, -2), (-2, 2), (2, ). We find
that the only interval in which 4 - x² is non-negative is (-2, 2).

We must exclude the endpoints -2, and +2 because the function would be 0
there, and no rectangle can have 0 area (unless we allow that a horizontal
line segment could be called "a rectangle with width 0" or that a
vertical line segment could be called "a rectangle with length 0").  So the
domain is

                 (-2, 2)

Edwin