Given the following graph of f(x), set up the approximation of the definite integral using the Trapezoidal Rule and n = 4. In your work, include your estimated y-values. Graph: -2 Integral: f(x)dx 2 2
Given the following graph of f(x), set up the approximation of the definite integral using the Trapezoidal Rule and n = 4. In your work, include your estimated y-values. Graph: -2 Integral: f(x)dx 2 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![## Approximating Definite Integrals Using the Trapezoidal Rule
### Problem Statement
Given the graph of \( f(x) \), set up the approximation of the definite integral using the Trapezoidal Rule with \( n = 4 \). In your work, include your estimated y-values.
### Integral to Approximate
\[
\int_{0}^{4} f(x) \, dx
\]
### Graph Explanation
The graph provided is a plot of the function \( f(x) \) over the interval \([0, 4]\). The x-axis ranges from approximately \(-4\) to \(4\), while the y-axis ranges from \(-4\) to \(8\). The curve indicates \( f(x) \) has both increasing and decreasing intervals, featuring a significant peak and trough within the specified range.
### Steps to Use the Trapezoidal Rule
1. **Divide the Interval**: The interval \([0, 4]\) is divided into \( n = 4 \) equal parts, each of width \( \Delta x = 1 \).
2. **Estimate y-values**: For each subinterval endpoint, estimate the y-value (height) from the graph.
3. **Apply the Trapezoidal Rule Formula**:
\[
\int_{0}^{4} f(x) \, dx \approx \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \right]
\]
where \( x_0, x_1, x_2, x_3, x_4 \) are the endpoints of the subintervals.
4. **Calculate the Approximation**: Insert the estimated y-values into the formula to find the approximate value of the integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d520b33-7440-45bc-bf50-9535ed2d03bc%2Fdecc7bdb-58f4-4abf-ae4c-7d793fc9aa05%2Fzunfiuoi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Approximating Definite Integrals Using the Trapezoidal Rule
### Problem Statement
Given the graph of \( f(x) \), set up the approximation of the definite integral using the Trapezoidal Rule with \( n = 4 \). In your work, include your estimated y-values.
### Integral to Approximate
\[
\int_{0}^{4} f(x) \, dx
\]
### Graph Explanation
The graph provided is a plot of the function \( f(x) \) over the interval \([0, 4]\). The x-axis ranges from approximately \(-4\) to \(4\), while the y-axis ranges from \(-4\) to \(8\). The curve indicates \( f(x) \) has both increasing and decreasing intervals, featuring a significant peak and trough within the specified range.
### Steps to Use the Trapezoidal Rule
1. **Divide the Interval**: The interval \([0, 4]\) is divided into \( n = 4 \) equal parts, each of width \( \Delta x = 1 \).
2. **Estimate y-values**: For each subinterval endpoint, estimate the y-value (height) from the graph.
3. **Apply the Trapezoidal Rule Formula**:
\[
\int_{0}^{4} f(x) \, dx \approx \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \right]
\]
where \( x_0, x_1, x_2, x_3, x_4 \) are the endpoints of the subintervals.
4. **Calculate the Approximation**: Insert the estimated y-values into the formula to find the approximate value of the integral.
Expert Solution
Step 1
From the given graph, it is observed that the value of .
The objective is to calculate the value of integration by the Trapezoidal rule for .
The value . Therefore, the number of subinterval is .
The initial point is and the final point is . Therefore, the length of each subinterval is .
The node points are .
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