ex f(x) = x2 + V1 + 2x, 4 s x < 6 lim n- co i = 1
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:**
Use this definition with right endpoints to find an expression for the area under the graph of \( f \) as a limit. Do not evaluate the limit.
\[ f(x) = x^2 + \sqrt{1 + 2x}, \quad 4 \leq x \leq 6 \]
\[ \lim_{n \to \infty} \sum_{i=1}^{n} \left( \text{(Expression in terms of } f(x_i) \text{ and } \Delta x) \right) \]
**Explanation:**
You are asked to find an expression for the area under the curve of the function \( f(x) = x^2 + \sqrt{1 + 2x} \) over the interval \([4, 6]\) using right endpoints. The limit expression represents the sum of the areas of rectangles under the curve as the number of rectangles \( n \) approaches infinity.
The notation involves:
- \( n \to \infty \) indicating an infinite number of small rectangles.
- \( \sum_{i=1}^{n} \) representing the summation of the areas of each rectangle from 1 to \( n \).
- The expression inside the summation will typically involve the width of each rectangle (\(\Delta x\)) and the function value at the right endpoint of each subinterval (\( f(x_i) \)).
Remember, you do not need to evaluate this limit; just express the area under \( f(x) \) in this limit form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc262d38-8642-4292-8f86-e10c24c29ddb%2Fe1c07b81-2553-474d-8296-d479eb203280%2Fucqc76j_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use this definition with right endpoints to find an expression for the area under the graph of \( f \) as a limit. Do not evaluate the limit.
\[ f(x) = x^2 + \sqrt{1 + 2x}, \quad 4 \leq x \leq 6 \]
\[ \lim_{n \to \infty} \sum_{i=1}^{n} \left( \text{(Expression in terms of } f(x_i) \text{ and } \Delta x) \right) \]
**Explanation:**
You are asked to find an expression for the area under the curve of the function \( f(x) = x^2 + \sqrt{1 + 2x} \) over the interval \([4, 6]\) using right endpoints. The limit expression represents the sum of the areas of rectangles under the curve as the number of rectangles \( n \) approaches infinity.
The notation involves:
- \( n \to \infty \) indicating an infinite number of small rectangles.
- \( \sum_{i=1}^{n} \) representing the summation of the areas of each rectangle from 1 to \( n \).
- The expression inside the summation will typically involve the width of each rectangle (\(\Delta x\)) and the function value at the right endpoint of each subinterval (\( f(x_i) \)).
Remember, you do not need to evaluate this limit; just express the area under \( f(x) \) in this limit form.
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