Answer
Two rectangles: $12$
Four rectangles: $11$
Work Step by Step
$ƒ(x) = 4 - x^{2}$ between x = -2 and x = 2 using the midpoint rule, we'll divide the interval [-2, 2] into rectangles and calculate the areas of those rectangles.
First, let's use two rectangles:
Divide the interval [-2, 2] into two equal subintervals: [-2, 0] and [0, 2].
Calculate the midpoints of the bases of the rectangles:
For the first rectangle: midpoint = $\frac{(-2 + 0)}{2}$= -1
For the second rectangle: midpoint = $\frac{(0 + 2)}{2}$= 1
Evaluate the function ƒ(x) = 4 - $x^{2}$ at the midpoints:
For the first rectangle: ƒ(-1) = 4 - $(-1)^{2}$= 4 - 1 = 3
For the second rectangle: ƒ(1) = 4 - $1^{2}$= 4 - 1 = 3
Calculate the areas of the rectangles:
For the first rectangle: area = 2 $\times$ 3 = 6
For the second rectangle: area = 2 $\times$ 3 = 6
Sum up the areas of the rectangles: total area ≈ 6 + 6 = 12
Now, let's use four rectangles:
Divide the interval [-2, 2] into four equal subintervals: [-2, -1], [-1, 0], [0, 1], and [1, 2].
Calculate the midpoints of the bases of the rectangles:
For the first rectangle: midpoint = $\frac{(-2 + -1)}{2}$= -1.5
For the second rectangle: midpoint = $\frac{(-1 + 0) }{2}$ = -0.5
For the third rectangle: midpoint = $\frac{(0 + 1)}{2}$= 0.5
For the fourth rectangle: midpoint = $\frac{(1 + 2)}{2}$= 1.5
Evaluate the function ƒ(x) = 4 - $x^{2}$ at the midpoints:
For the first rectangle: ƒ(-1.5) = 4 - $(-1.5)^{2}$= 4 - 2.25 = 1.75
For the second rectangle: ƒ(-0.5) = 4 - $(-0.5)^{2}$= 4 - 0.25 = 3.75
For the third rectangle: ƒ(0.5) = 4 - $(0.5)^{2}$= 4 - 0.25 = 3.75
For the fourth rectangle: ƒ(1.5) = 4 - $(1.5)^{2}$= 4 - 2.25 = 1.75
Calculate the areas of the rectangles:
For each rectangle: area = 1 $\times$ ƒ(midpoint)
For the first rectangle: area = 1 $\times$ 1.75 = 1.75
For the second rectangle: area = 1 $\times$ 3.75 = 3.75
For the third rectangle: area = 1 $\times$ 3.75 = 3.75
For the fourth rectangle: area = 1 $\times$ 1.75 = 1.75
Sum up the areas of the rectangles: total area ≈ 1.75 + 3.75 + 3.75 + 1.75 = 11
