Answer
Two rectangles: $0.3125$
Four rectangles: $0.328125$
Work Step by Step
Given, $ƒ(x) = x^2$ between $x = 0$ and $x = 1$, using the midpoint rule, we can divide the interval $[0, 1]$ into rectangles and calculate the area of each rectangle.
First, let's start with two rectangles:
Divide the interval $[0, 1]$ into two equal subintervals:
Subinterval $1$: $[0, 0.5]$
Subinterval $2$: $[0.5, 1]$
Find the midpoint of each subinterval:
Midpoint $1$:$ x = \frac{(0 + 0.5)}{2}= 0.25$
Midpoint $2$: $x = \frac{(0.5 + 1) }{2} = 0.75$
Calculate the height of each rectangle using the function $ƒ(x) = x^{2}$ at the corresponding midpoints:
Height $1$: $ƒ(0.25) = 0.25^{2}= 0.0625$
Height $2$: $ƒ(0.75) = 0.75^{2}= 0.5625$
Calculate the width of each rectangle:
Width $1$: $0.5$ (since the subintervals have equal width)
Width $2$: $0.5$
Calculate the area of each rectangle:
Area $1$: Height $1$ $\times$ Width $1 = 0.0625 \times 0.5 = 0.03125$
Area $2$: Height $2$ $\times$ Width $2 = 0.5625 \times 0.5 = 0.28125$
Estimate the total area under the graph by summing the areas of the rectangles:
Total area = Area 1 + Area 2$ = 0.03125 + 0.28125 = 0.3125$
Therefore, using two rectangles, the estimated area under the graph of $ƒ(x) = x^{2}$ between $x = 0$ and $x = 1$ is approximately $0.3125$.
Now let's repeat the process using four rectangles:
Divide the interval [0, 1] into four equal subintervals:
Subinterval 1: [0, 0.25]
Subinterval 2: [0.25, 0.5]
Subinterval 3: [0.5, 0.75]
Subinterval 4: [0.75, 1]
Find the midpoint of each subinterval:
Midpoint 1: x = $\frac{(0 + 0.25)}{2}$ = 0.125
Midpoint 2: x = $\frac{(0.25 + 0.5) }{2}$= 0.375
Midpoint 3: x = $\frac{(0.5 + 0.75)}{2}$= 0.625
Midpoint 4: x = $\frac{(0.75 + 1)}{2}$= 0.875
Calculate the height of each rectangle using the function ƒ(x) = x^2 at the corresponding midpoints:
Height 1: ƒ(0.125) = $(0.125)^{2}$ = 0.015625
Height 2: ƒ(0.375) = $(0.375)^{2}$= 0.140625
Height 3: ƒ(0.625) = $(0.625)^{2}$= 0.390625
Height 4: ƒ(0.875) = $(0.875)^{2}$ = 0.765625
Calculate the width of each rectangle:
Width 1: 0.25 (since the subintervals have equal width)
Width 2: 0.25
Width 3: 0.25
Width 4: 0.25
Calculate the area of each rectangle:
Area 1: Height 1 $\times$ Width 1 = 0.015625 $\times$ 0.25 = 0.00390625
Area 2: Height 2 $\times$ Width 2 = 0.140625 $\times$ 0.25 = 0.03515625
Area 3: Height 3 $\times$ Width 3 = 0.390625 $\times$ 0.25 = 0.09765625
Area 4: Height 4 $\times$ Width 4 = 0.765625 $\times$ 0.25 = 0.19140625
Estimate the total area under the graph by summing the areas of the rectangles:
Total area = Area 1 + Area 2 + Area 3 + Area 4 = 0.00390625 + 0.03515625 + 0.09765625 + 0.19140625 = 0.328125
Therefore, using four rectangles, the estimated area under the graph of ƒ(x) = $x^{2}$ between x = 0 and x = 1 is approximately 0.328125.
