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
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**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}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1df78ed7-7bf9-40d0-9ea6-9fdf014e5eeb%2F51713113-9810-443f-8559-b9a5f8c999ff%2Fcwfx87iq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**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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