e Given the graph of the function f (x) below with 4 subintervals, write down expressions in terms of f(x) for each of the five integral approximation methods for | f(x)dx .
e Given the graph of the function f (x) below with 4 subintervals, write down expressions in terms of f(x) for each of the five integral approximation methods for | f(x)dx .
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
![### Integral Approximation of \( f(x) \)
Given the graph of the function \( f(x) \) below with 4 subintervals, the task is to write expressions in terms of \( f(x) \) for each of the five integral approximation methods for
\[
\int_{x_0}^{x_4} f(x) \, dx .
\]
#### Approximation Methods
- **\( L_4 \) (Left Riemann Sum)**
- **\( R_4 \) (Right Riemann Sum)**
- **\( M_4 \) (Midpoint Riemann Sum)**
- **\( T_4 \) (Trapezoidal Rule)**
- **\( S_4 \) (Simpson’s Rule)**
#### Task:
- List the following in order from smallest to largest:
\[
L_4, R_4, M_4, T_4, S_4, \int_{x_0}^{x_4} f(x) \, dx .
\]
#### Instructions:
1. Re-read Example 2 carefully.
2. Repeat the calculations using \(\int_{0}^{2} x^3 \, dx\) instead of \(\int_{1}^{2} \frac{1}{x} \, dx\).
- **Show your work on the back of this sheet.**
#### Reflection Questions:
- How much time did you spend reading this section and completing this assignment? ____ minutes.
- What questions do you have after reading this section? (Use the back if necessary.)
### Graph Explanation:
The graph displays a curve representing the function \( f(x) \) from \( x_0 \) to \( x_4 \). It is divided into four subintervals. Each subinterval is marked by vertical lines that connect the x-axis to the curve. This visual representation assists in understanding how each approximation method estimates the area under the curve.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F953e3ae3-1a36-4eca-a542-ddeb8c6ae72a%2F0b9f85dd-0409-429c-a068-a853825c0e39%2F8dfoiyqk_processed.png&w=3840&q=75)
Transcribed Image Text:### Integral Approximation of \( f(x) \)
Given the graph of the function \( f(x) \) below with 4 subintervals, the task is to write expressions in terms of \( f(x) \) for each of the five integral approximation methods for
\[
\int_{x_0}^{x_4} f(x) \, dx .
\]
#### Approximation Methods
- **\( L_4 \) (Left Riemann Sum)**
- **\( R_4 \) (Right Riemann Sum)**
- **\( M_4 \) (Midpoint Riemann Sum)**
- **\( T_4 \) (Trapezoidal Rule)**
- **\( S_4 \) (Simpson’s Rule)**
#### Task:
- List the following in order from smallest to largest:
\[
L_4, R_4, M_4, T_4, S_4, \int_{x_0}^{x_4} f(x) \, dx .
\]
#### Instructions:
1. Re-read Example 2 carefully.
2. Repeat the calculations using \(\int_{0}^{2} x^3 \, dx\) instead of \(\int_{1}^{2} \frac{1}{x} \, dx\).
- **Show your work on the back of this sheet.**
#### Reflection Questions:
- How much time did you spend reading this section and completing this assignment? ____ minutes.
- What questions do you have after reading this section? (Use the back if necessary.)
### Graph Explanation:
The graph displays a curve representing the function \( f(x) \) from \( x_0 \) to \( x_4 \). It is divided into four subintervals. Each subinterval is marked by vertical lines that connect the x-axis to the curve. This visual representation assists in understanding how each approximation method estimates the area under the curve.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 5 steps with 1 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
