Answer
Two rectangles: $0.21875$
Four rectangles: $0.24599511719$
Work Step by Step
Given-ƒ(x) = $x^{3}$ between x = 0 and x = 1, using the midpoint rule, we'll divide the interval [0, 1] into rectangles and calculate the areas of those rectangles.
First, let's use two rectangles:
Divide the interval [0, 1] into two equal subintervals: [0, 0.5] and [0.5, 1].
Calculate the midpoints of the bases of the rectangles:
For the first rectangle: midpoint = $\frac{(0 + 0.5)}{2}$= 0.25
For the second rectangle: midpoint = $\frac{(0.5 + 1)}{2}$= 0.75
Evaluate the function ƒ(x) = $x^{3}$ at the midpoints:
For the first rectangle: ƒ(0.25) = $0.25^{3}$ = 0.015625
For the second rectangle: ƒ(0.75) = $0.75^{3}$ = 0.421875
Calculate the areas of the rectangles:
For the first rectangle: area = 0.5 $\times$ 0.015625 = 0.0078125
For the second rectangle: area = 0.5 $\times$ 0.421875 = 0.2109375
Sum up the areas of the rectangles: total area ≈ 0.0078125 + 0.2109375 ≈ 0.21875
Now, let's use four rectangles:
Divide the interval [0, 1] into four equal subintervals: [0, 0.25], [0.25, 0.5], [0.5, 0.75], and [0.75, 1].
Calculate the midpoints of the bases of the rectangles:
For the first rectangle: midpoint = $\frac{(0 + 0.25)}{2}$= 0.125
For the second rectangle: midpoint = $\frac{(0.25 + 0.5) }{2}$ = 0.375
For the third rectangle: midpoint = $\frac{(0.5 + 0.75) }{2}$ = 0.625
For the fourth rectangle: midpoint = $\frac{(0.75 + 1)}{2}$= 0.875
Evaluate the function ƒ(x) = $x^{3}$ at the midpoints:
For the first rectangle: ƒ(0.125) = $0.125^{3}$= 0.001953125
For the second rectangle: ƒ(0.375) = $0.375^{3}$= 0.052734375
For the third rectangle: ƒ(0.625) = $0.625^{3}$= 0.244140625
For the fourth rectangle: ƒ(0.875) = $0.875^{3}$ = 0.681152344
Calculate the areas of the rectangles:
For each rectangle: area = 0.25 $\times$ ƒ(midpoint)
For the first rectangle: area = 0.25 $\times$ 0.001953125 = 0.00048828125
For the second rectangle: area = 0.25 $\times$ 0.052734375 = 0.01318359375
For the third rectangle: area = 0.25 $\times$ 0.244140625 = 0.06103515625
For the fourth rectangle: area = 0.25 $\times$ 0.681152344 = 0.17028808594
Sum up the areas of the rectangles: total area ≈ 0.00048828125 + 0.01318359375 + 0.06103515625 + 0.17028808594 ≈ 0.24599511719
