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]?

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]?

• A. 8
• B. 10
• C. 12
• D. 16

Answer: (B)

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 +

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 +