Using four right endpoint rectangles of equal width, what is the approximate area under the curve f(x) = -x² + 4 over the interval [-2, 2]?

To find the approximate area under the curve ( f(x) = -x^2 + 4 ) over the interval ([-2, 2]) using four right endpoint rectangles, we first determine the width of each rectangle. The interval length is (2 - (-2) = 4), and dividing this into four equal segments results in a width of (1) for each rectangle.

The right endpoints of these rectangles, based on the width, occur at the points:

• (x = -1)

• (x = 0)

• (x = 1)

• (x = 2)

Next, we calculate the height of the function at each right endpoint:

1. For (x = -1):

[

f(-1) = -(-1)^2 + 4 = -1 + 4 = 3

]

2. For (x = 0):

[

f(0) = -(0)^2 + 4 = 0 + 4 = 4

]

3. For (x = 1):

[

f(1) = -(1)^2 +