A rectangle is bounded by the x-axis and the semicircle Y= SQ RT of 4-x^2. Write the area A of the rectangle as a function of x and determine the domain of the function.

In google type this question and you will see answer

To find the area, A, of the rectangle bounded by the x-axis and the semicircle y = √(4 - x^2), we need to determine the dimensions of the rectangle. The base of the rectangle will be the distance between the x-intercepts of the semicircle, and the height will be the y-coordinate of the semicircle at that x-intercept.

Let's start by finding the x-intercepts of the semicircle:

Setting y = 0 in the equation of the semicircle, we have:
0 = √(4 - x^2)

Squaring both sides of the equation, we get:
0 = 4 - x^2

Rearranging the equation, we have:
x^2 = 4

Taking the square root of both sides, we obtain:
x = ±√4

So, the x-intercepts are x = -2 and x = 2.

Since the rectangle is bounded by the x-axis, the y-coordinate at each x-intercept will be 0. Therefore, the height of the rectangle is 0.

Now let's find the base of the rectangle, which is the distance between the x-intercepts:

Base = 2 - (-2)
Base = 2 + 2
Base = 4

Since the height is 0, the area of the rectangle, A, will be:

A = Base * Height
A = 4 * 0
A = 0

Therefore, the area of the rectangle is zero.

As for the domain of the function, we consider the x-values for which the function y = √(4 - x^2) is defined. The function is defined when the expression inside the square root is greater than or equal to zero:

4 - x^2 ≥ 0

Rearranging the inequality, we have:
x^2 ≤ 4

Taking the square root of both sides, we get:
|x| ≤ 2

This means that the values of x must be within the interval [-2, 2] for the function to be defined. Thus, the domain of the function is [-2, 2].