A rectangle is bounded by the x-axis and the semicircle y = ¡Ì36 ¨C 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 = 2x*sqrt(36-x^2)
Domain would be -6<=x<=6
To find the area of the rectangle, we need to determine the length of its sides.
First, let's look at the figure. We have a rectangle bounded by the x-axis and a semicircle with equation y = √(36 - x^2).
The rectangle's base is parallel to the x-axis and its height corresponds to the difference in y-values between the x-axis and the semicircle.
The base of the rectangle is the same as the x-axis, so its length is simply x.
To find the height, we need to compute the difference between the y-coordinate of the semicircle at x and the value of the x-axis at x.
The y-coordinate of the semicircle at x is given by y = √(36 - x^2).
The value of the x-axis at x is simply 0.
Therefore, the height of the rectangle is √(36 - x^2) - 0 = √(36 - x^2).
Now that we have the length and height of the rectangle, we can find its area by multiplying these two values together:
A(x) = length × height = x × √(36 - x^2).
The domain of the area function is determined by the restrictions on x imposed by the problem. In this case, x can vary from the leftmost point of the semicircle to the rightmost point of the semicircle.
The equation of the semicircle is y = √(36 - x^2). To find the leftmost and rightmost points, we set y = 0:
0 = √(36 - x^2)
0 = 36 - x^2
x^2 = 36
x = ±√36
x = ±6
Therefore, the domain of the area function is -6 ≤ x ≤ 6, or in interval notation, [-6, 6].