Evaluate the Riemann sum for f(x) = 3x - 1,-6 ≤ x ≤ 4, with five subintervals, taking the sample points to be right endpoints. 0 Explain, with the aid of a diagram, what the Riemann sum represents.
Evaluate the Riemann sum for f(x) = 3x - 1,-6 ≤ x ≤ 4, with five subintervals, taking the sample points to be right endpoints. 0 Explain, with the aid of a diagram, what the Riemann sum represents.
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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![**Problem Statement:**
Evaluate the Riemann sum for \( f(x) = 3x - 1 \), over the interval \(-6 \leq x \leq 4\), using five subintervals, and taking the sample points to be right endpoints.
**User Input:**
A box with the entry "0" marked with a red "X," indicating an incorrect answer.
**Task:**
Explain, with the aid of a diagram, what the Riemann sum represents.
---
**Explanation:**
A Riemann sum is a way to approximate the total area under a curve on a graph, otherwise known as the integral of a function. It works by dividing the area into simple shapes (rectangles or trapezoids), calculating the area for each shape, and then summing these areas.
For the function \( f(x) = 3x - 1 \), the graph is a straight line. The interval \([-6, 4]\) is divided into five subintervals. For each subinterval, a rectangle is drawn with a height equal to the value of the function at the right endpoint of the subinterval. The width of each rectangle is the length of each subinterval.
**Steps to Calculate the Riemann Sum:**
1. **Divide the interval \([-6, 4]\) into five equal parts:**
Each subinterval has a length of \(\frac{4 - (-6)}{5} = 2\).
2. **Determine the right endpoints:**
The right endpoints are \(-4, -2, 0, 2, 4\).
3. **Evaluate the function \( f(x) = 3x - 1 \) at each right endpoint:**
- \( f(-4) = 3(-4) - 1 = -13 \)
- \( f(-2) = 3(-2) - 1 = -7 \)
- \( f(0) = 3(0) - 1 = -1 \)
- \( f(2) = 3(2) - 1 = 5 \)
- \( f(4) = 3(4) - 1 = 11 \)
4. **Calculate the area of each rectangle and sum them:**
\[
\text{Riemann Sum} = (2 \times -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61b74698-3f34-41c6-b209-082ec83323f3%2F964c81db-9517-409d-9af0-79cad7afb1b5%2Fbl7c8uu_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Evaluate the Riemann sum for \( f(x) = 3x - 1 \), over the interval \(-6 \leq x \leq 4\), using five subintervals, and taking the sample points to be right endpoints.
**User Input:**
A box with the entry "0" marked with a red "X," indicating an incorrect answer.
**Task:**
Explain, with the aid of a diagram, what the Riemann sum represents.
---
**Explanation:**
A Riemann sum is a way to approximate the total area under a curve on a graph, otherwise known as the integral of a function. It works by dividing the area into simple shapes (rectangles or trapezoids), calculating the area for each shape, and then summing these areas.
For the function \( f(x) = 3x - 1 \), the graph is a straight line. The interval \([-6, 4]\) is divided into five subintervals. For each subinterval, a rectangle is drawn with a height equal to the value of the function at the right endpoint of the subinterval. The width of each rectangle is the length of each subinterval.
**Steps to Calculate the Riemann Sum:**
1. **Divide the interval \([-6, 4]\) into five equal parts:**
Each subinterval has a length of \(\frac{4 - (-6)}{5} = 2\).
2. **Determine the right endpoints:**
The right endpoints are \(-4, -2, 0, 2, 4\).
3. **Evaluate the function \( f(x) = 3x - 1 \) at each right endpoint:**
- \( f(-4) = 3(-4) - 1 = -13 \)
- \( f(-2) = 3(-2) - 1 = -7 \)
- \( f(0) = 3(0) - 1 = -1 \)
- \( f(2) = 3(2) - 1 = 5 \)
- \( f(4) = 3(4) - 1 = 11 \)
4. **Calculate the area of each rectangle and sum them:**
\[
\text{Riemann Sum} = (2 \times -
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