**Understanding Riemann Sums and Area Under the Curve**For the function given below, find a formula for the Riemann sum obtained by dividing the interval $[a,b]$ into n equal subintervals and using the right-hand endpoint for each $n$. Then take a limit of this sum as $n \rightarrow \infty$ to calculate the area under the curve over $[a,b]$.Given Function:\[ f(x) = 4x \] over the interval $[2, 8]$.### Step-by-Step Process1. **Define the Subintervals**: - Divide the interval $[2, 8]$ into $n$ equal subintervals. - Each subinterval has a width $\Delta x = \frac{8 - 2}{n} = \frac{6}{n}$.2. **Identify the Right-End Points**: - The right-end points of the subintervals can be defined as: \[ x_i = 2 + i \cdot \Delta x = 2 + i \cdot \frac{6}{n} \] where $i=1, 2, ..., n$.3. **Set Up the Riemann Sum**: - Using the right-end points, the Riemann sum $S_n$ is: \[ S_n = \sum_{i=1}^n f(x_i) \cdot \Delta x = \sum_{i=1}^n 4 \left( 2 + i \cdot \frac{6}{n} \right) \cdot \frac{6}{n} \]4. **Simplify the Sum**: \[ S_n = 4 \cdot \frac{6}{n} \sum_{i=1}^n \left( 2 + i \cdot \frac{6}{n} \right) = \frac{24}{n} \sum_{i=1}^n \left( 2 + \frac{6i}{n} \right) \] Distribute and split the sum: \[ S_n = \frac{24}{n} \sum_{i=1}^n 2 + \frac{24}{n}

Definite integral expressed as Riemann sum. where and . Note .

For the function given below, find a formula for the Riemann sum obtained by dividing the interval (a.b] into n equal subintervals and using the right-hand endpoint for eacho,. Then take a limit of this sum as n+ o to calculate the area under the curve over [a,b). f(x) = 4x over the interval [2,6). Find a formula for the Riemann sum. The area under the curve over [2,8] is square units. (Simplify your answer.)

For the function given below, find a formula for the Riemann sum obtained by dividing the interval (a.b] into n equal subintervals and using the right-hand endpoint for eacho,. Then take a limit of this sum as n+ o to calculate the area under the curve over [a,b). f(x) = 4x over the interval [2,6). Find a formula for the Riemann sum. The area under the curve over [2,8] is square units. (Simplify your answer.)

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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Question

**Understanding Riemann Sums and Area Under the Curve**

For the function given below, find a formula for the Riemann sum obtained by dividing the interval $[a,b]$ into n equal subintervals and using the right-hand endpoint for each $n$. Then take a limit of this sum as $n \rightarrow \infty$ to calculate the area under the curve over $[a,b]$.

Given Function:
\[ f(x) = 4x \]
over the interval $[2, 8]$.

### Step-by-Step Process

1. **Define the Subintervals**:
- Divide the interval $[2, 8]$ into $n$ equal subintervals.
- Each subinterval has a width $\Delta x = \frac{8 - 2}{n} = \frac{6}{n}$.

2. **Identify the Right-End Points**:
- The right-end points of the subintervals can be defined as:
\[
x_i = 2 + i \cdot \Delta x = 2 + i \cdot \frac{6}{n}
\]
where $i=1, 2, ..., n$.

3. **Set Up the Riemann Sum**:
- Using the right-end points, the Riemann sum $S_n$ is:
\[
S_n = \sum_{i=1}^n f(x_i) \cdot \Delta x
= \sum_{i=1}^n 4 \left( 2 + i \cdot \frac{6}{n} \right) \cdot \frac{6}{n}
\]

4. **Simplify the Sum**:
\[
S_n = 4 \cdot \frac{6}{n} \sum_{i=1}^n \left( 2 + i \cdot \frac{6}{n} \right)
= \frac{24}{n} \sum_{i=1}^n \left( 2 + \frac{6i}{n} \right)
\]
Distribute and split the sum:
\[
S_n = \frac{24}{n} \sum_{i=1}^n 2 + \frac{24}{n}

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