Use right and left endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval. f(x) = 2x + 2, [0, 2], 4 rectangles Step 1 To find two approximations of the area of the region between the graph of the function f(x) = 2x + 2 and the x-axis over the interval [0, 2] using 4 rectangles, first partition the interval [0, 2 ] into n = | Then the width of each rectangle is given by using the following formula. b-a Ax= Step 2 Therefore, the width of each rectangle is b-a n Ax = n 4 -0.5 2 - 0 Step 3 Consider the right endpoints approximation of the area of the region. Observe the region between the graph of the function f(x) = 2x + 2 and the x-axis over the interval [0, 2] with four circumscribed rectangles, which is shown below. 1 6 4 0.5 1.0 1.5 The right endpoints of the n intervals are Ax(i) where i = 1 to 4 Thus, the right end points of four intervals are Ax(i) = 2 2.0 Then substitute 4x ==-- and n = 4 to find the left end points of four intervals. 1 That is, the four right end points of the intervals are Therefore, the four intervals are given as follows. ·[글·ㄷ 2 2.5 1 (i), wherei 1 to 4. 1, 2 2 X and 2. (222) 4 subintervals.
Use right and left endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval. f(x) = 2x + 2, [0, 2], 4 rectangles Step 1 To find two approximations of the area of the region between the graph of the function f(x) = 2x + 2 and the x-axis over the interval [0, 2] using 4 rectangles, first partition the interval [0, 2 ] into n = | Then the width of each rectangle is given by using the following formula. b-a Ax= Step 2 Therefore, the width of each rectangle is b-a n Ax = n 4 -0.5 2 - 0 Step 3 Consider the right endpoints approximation of the area of the region. Observe the region between the graph of the function f(x) = 2x + 2 and the x-axis over the interval [0, 2] with four circumscribed rectangles, which is shown below. 1 6 4 0.5 1.0 1.5 The right endpoints of the n intervals are Ax(i) where i = 1 to 4 Thus, the right end points of four intervals are Ax(i) = 2 2.0 Then substitute 4x ==-- and n = 4 to find the left end points of four intervals. 1 That is, the four right end points of the intervals are Therefore, the four intervals are given as follows. ·[글·ㄷ 2 2.5 1 (i), wherei 1 to 4. 1, 2 2 X and 2. (222) 4 subintervals.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Certainly! Here is the transcription of the provided image, formatted as it might appear on an educational website:
---
**Title: Approximating the Area Under a Curve Using Right and Left Endpoints**
**Objective:** Use right and left endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval.
**Function:** \( f(x) = 2x + 2 \)
**Interval:** \([0, 2]\)
**Number of Rectangles:** 4
---
### Step 1
To find two approximations of the area of the region between the graph of the function \( f(x) = 2x + 2 \) and the x-axis over the interval \([0, 2]\) using 4 rectangles, first partition the interval \([0, 2]\) into \( n = 4 \) subintervals.
Then the width of each rectangle is given by using the following formula:
\[
\Delta x = \frac{b-a}{n}
\]
### Step 2
Therefore, the width of each rectangle is:
\[
\Delta x = \frac{b-a}{n} = \frac{2 - 0}{4} = \frac{1}{2}
\]
### Step 3
Consider the right endpoints approximation of the area of the region.
Observe the region between the graph of the function \( f(x) = 2x + 2 \) and the x-axis over the interval \([0, 2]\) with four circumscribed rectangles, as shown below in the diagram.
**Diagram Description:**
In the provided graph, there is a line representing the function \( f(x) \). There are four rectangles with their right endpoints touching the line, indicating the method of approximation for the area under the curve.
The right endpoints of the \( n \) intervals are \(\Delta x(i)\) where \( i = 1 \) to 4.
Then substitute \(\Delta x = \frac{1}{2}\) and \( n = 4 \) to find the left end points of four intervals.
Thus, the right end points of four intervals are \(\Delta x(i) = \frac{1}{2}i\), where \( i = 1 \) to 4.
That is, the four right](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9b6a03f-2298-4441-ab19-fb50f28ec42f%2F104747fc-8d79-4720-a78e-ee28d91dda62%2Fpk8tacu_processed.png&w=3840&q=75)
Transcribed Image Text:Certainly! Here is the transcription of the provided image, formatted as it might appear on an educational website:
---
**Title: Approximating the Area Under a Curve Using Right and Left Endpoints**
**Objective:** Use right and left endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval.
**Function:** \( f(x) = 2x + 2 \)
**Interval:** \([0, 2]\)
**Number of Rectangles:** 4
---
### Step 1
To find two approximations of the area of the region between the graph of the function \( f(x) = 2x + 2 \) and the x-axis over the interval \([0, 2]\) using 4 rectangles, first partition the interval \([0, 2]\) into \( n = 4 \) subintervals.
Then the width of each rectangle is given by using the following formula:
\[
\Delta x = \frac{b-a}{n}
\]
### Step 2
Therefore, the width of each rectangle is:
\[
\Delta x = \frac{b-a}{n} = \frac{2 - 0}{4} = \frac{1}{2}
\]
### Step 3
Consider the right endpoints approximation of the area of the region.
Observe the region between the graph of the function \( f(x) = 2x + 2 \) and the x-axis over the interval \([0, 2]\) with four circumscribed rectangles, as shown below in the diagram.
**Diagram Description:**
In the provided graph, there is a line representing the function \( f(x) \). There are four rectangles with their right endpoints touching the line, indicating the method of approximation for the area under the curve.
The right endpoints of the \( n \) intervals are \(\Delta x(i)\) where \( i = 1 \) to 4.
Then substitute \(\Delta x = \frac{1}{2}\) and \( n = 4 \) to find the left end points of four intervals.
Thus, the right end points of four intervals are \(\Delta x(i) = \frac{1}{2}i\), where \( i = 1 \) to 4.
That is, the four right
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Follow-up Question
Σ[(21)](3) - Σ | 2
4
-
4
=
-Σ||
i=1
=
1
Step 10
Therefore, using the left end points, the sum of the areas of four rectangles is
¡- 1
KB)
i=1
2
- (-/-)
= 5 +
2
+
4
2
i=1
(¹=¹)
2
1
to approximate the sum of area using the left end points.
1
2
3
NH
1, and
+2
1), wherei = 1 to 4.
3
+ 1)
2
+
2
3
3
(¹)
4](https://content.bartleby.com/qna-images/question/f9b6a03f-2298-4441-ab19-fb50f28ec42f/c264455c-9669-4730-9949-f912f35b7881/ctnopv_processed.png)
Transcribed Image Text:Thus, the left end points of four intervals are Ax(i – 1) =
That is, the four left end points of the intervals are 0,
Therefore, the four intervals are given as follows.
where i = 1 to 4.
Step 9
Using the left end points the height of each rectangle is
1
The width of each rectangle is
4
0
i=1
-
4
i=1
1
1
Step 11
simplify Σ{f(/z1)](3)
Σ[(21)](3) - Σ | 2
4
-
4
=
-Σ||
i=1
=
1
Step 10
Therefore, using the left end points, the sum of the areas of four rectangles is
¡- 1
KB)
i=1
2
- (-/-)
= 5 +
2
+
4
2
i=1
(¹=¹)
2
1
to approximate the sum of area using the left end points.
1
2
3
NH
1, and
+2
1), wherei = 1 to 4.
3
+ 1)
2
+
2
3
3
(¹)
4
Solution
by Bartleby ExpertFollow-up Question
i=1
i +2 (²²)
Step 7
i=1
4
i=1
=
-0.5
to approximate the sum of the area using right end points.
=
=
Then substitute Ax =
i=1
1
6
= 5+
5
i=1
NH
S
+
i + 2
i=1
0.5
4
Thus, the approximate sum of the area using right end points is 9
Step 8
Consider the left endpoints approximation of the area of the region.
2
2
Observe the region between the graph of the function f(x) = 2x + 2 and the x-axis over the interval [0, 2
Submit Skip (you cannot come back)
1.0
+ 1)
1.5
The left endpoints of the n intervals are Ax(i − 1) where i = 1 to
+
Thus, the left end points of four intervals are Ax(i – 1)
That is, the four left end points of the intervals are 0,
Therefore, the four intervals are given as follows.
2.0
11/13
and n = 4 to find the left end points of four intervals.
2
]], [[
2
2
1
9
2.5
4
(i − 1), wherei = 1 to 4.
1, and
2
2
2
2]
with four inscribed rectangles shown in the following figure.](https://content.bartleby.com/qna-images/question/f9b6a03f-2298-4441-ab19-fb50f28ec42f/c4eeeca2-6701-47b4-9f7f-135ecff8537e/e676kge_processed.png)
Transcribed Image Text:Step 6
Simplify
(16)
4
Σ{(J) -Σ[• L®2 (2)+ E2](;)
i=1
i +2 (²²)
Step 7
i=1
4
i=1
=
-0.5
to approximate the sum of the area using right end points.
=
=
Then substitute Ax =
i=1
1
6
= 5+
5
i=1
NH
S
+
i + 2
i=1
0.5
4
Thus, the approximate sum of the area using right end points is 9
Step 8
Consider the left endpoints approximation of the area of the region.
2
2
Observe the region between the graph of the function f(x) = 2x + 2 and the x-axis over the interval [0, 2
Submit Skip (you cannot come back)
1.0
+ 1)
1.5
The left endpoints of the n intervals are Ax(i − 1) where i = 1 to
+
Thus, the left end points of four intervals are Ax(i – 1)
That is, the four left end points of the intervals are 0,
Therefore, the four intervals are given as follows.
2.0
11/13
and n = 4 to find the left end points of four intervals.
2
]], [[
2
2
1
9
2.5
4
(i − 1), wherei = 1 to 4.
1, and
2
2
2
2]
with four inscribed rectangles shown in the following figure.
Solution
by Bartleby ExpertFollow-up Question
![### Detailed Explanation for Education
#### Calculating Right End Points
1. **Initial Substitution**
Substitute \(\Delta x = \frac{1}{2}\) and \(n = 4\) to determine the right end points of four intervals.
2. **Determine Right End Points**
The right end points of four intervals are given by:
\[
x(i) = \frac{1}{2} \cdot (i), \text{ where } i = 1 \text{ to } 4.
\]
Thus, the four right end points of the intervals are \(\frac{1}{2}, 1, \frac{3}{2}, 2\).
3. **Intervals**
The intervals, based on right end points, are:
\[
[0, \frac{1}{2}], [\frac{1}{2}, 1], [1, \frac{3}{2}], [\frac{3}{2}, 2].
\]
#### Step 4: Height of Rectangles
- Using the right end points, the height of each rectangle is calculated using:
\[
\left(\frac{2}{x(i)}\right), \text{ where } i = 1 \text{ to } 4.
\]
- The width of each rectangle is \(\frac{1}{2}\).
#### Step 5: Sum of Areas of Rectangles
- Using the right end points, the sum of the areas of four rectangles is:
\[
\sum_{i=1}^{4} \left(\frac{2}{x(i)}\right) \cdot \frac{1}{2}.
\]
#### Step 6: Simplifying the Sum
To simplify \(\sum_{i=1}^{4} \left(\frac{2}{x(i)}\right) \cdot \frac{1}{2}\):
1. Break down the expression:
\[
\sum_{i=1}^{4} \left(\frac{2}{x(i)}\right) \cdot \frac{1}{2} = \sum_{i=1}^{4} \left(\frac{2}{\frac{i}{2}}\right) \cdot \frac{1}{2}.
\]
2. Simplification inside the sum:
\[](https://content.bartleby.com/qna-images/question/f9b6a03f-2298-4441-ab19-fb50f28ec42f/4feecbfa-cf81-446a-9e90-85108afd4440/1edumla_processed.png)
Transcribed Image Text:### Detailed Explanation for Education
#### Calculating Right End Points
1. **Initial Substitution**
Substitute \(\Delta x = \frac{1}{2}\) and \(n = 4\) to determine the right end points of four intervals.
2. **Determine Right End Points**
The right end points of four intervals are given by:
\[
x(i) = \frac{1}{2} \cdot (i), \text{ where } i = 1 \text{ to } 4.
\]
Thus, the four right end points of the intervals are \(\frac{1}{2}, 1, \frac{3}{2}, 2\).
3. **Intervals**
The intervals, based on right end points, are:
\[
[0, \frac{1}{2}], [\frac{1}{2}, 1], [1, \frac{3}{2}], [\frac{3}{2}, 2].
\]
#### Step 4: Height of Rectangles
- Using the right end points, the height of each rectangle is calculated using:
\[
\left(\frac{2}{x(i)}\right), \text{ where } i = 1 \text{ to } 4.
\]
- The width of each rectangle is \(\frac{1}{2}\).
#### Step 5: Sum of Areas of Rectangles
- Using the right end points, the sum of the areas of four rectangles is:
\[
\sum_{i=1}^{4} \left(\frac{2}{x(i)}\right) \cdot \frac{1}{2}.
\]
#### Step 6: Simplifying the Sum
To simplify \(\sum_{i=1}^{4} \left(\frac{2}{x(i)}\right) \cdot \frac{1}{2}\):
1. Break down the expression:
\[
\sum_{i=1}^{4} \left(\frac{2}{x(i)}\right) \cdot \frac{1}{2} = \sum_{i=1}^{4} \left(\frac{2}{\frac{i}{2}}\right) \cdot \frac{1}{2}.
\]
2. Simplification inside the sum:
\[
Solution
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