Determine a region whose area is equal to the given limit. Do not evaluate the limit. 8i = √ ₁ + ²/ 1 lim コ→g O√1+x on [-8, 8] O 8√1+x on [-8, 8] O 8√1-x on [0, 8] O 8√1+x on [0, 8] O√1+x on [0, 8]
Determine a region whose area is equal to the given limit. Do not evaluate the limit. 8i = √ ₁ + ²/ 1 lim コ→g O√1+x on [-8, 8] O 8√1+x on [-8, 8] O 8√1-x on [0, 8] O 8√1+x on [0, 8] O√1+x on [0, 8]
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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![### Calculus: Evaluating Limits and Areas
#### Problem Statement:
Determine a region whose area is equal to the given limit. Do not evaluate the limit.
\[
\lim_{n \to \infty} \sum_{i=1}^{n} \frac{8}{n} \sqrt{1 + \frac{8i}{n}}
\]
#### Multiple Choice Options:
1. \(\sqrt{1 + x} \text{ on } [-8, 8]\)
2. \(8\sqrt{1 + x} \text{ on } [-8, 8]\)
3. \(8\sqrt{1 - x} \text{ on } [0, 8]\)
4. \(8\sqrt{1 + x} \text{ on } [0, 8]\)
5. \(\sqrt{1 + x} \text{ on } [0, 8]\)
### Solution:
Analyze the given limit and identify the region whose area it represents by interpreting the sum as a Riemann sum. The Riemann sum approximation for the area under the curve \( y = \sqrt{1 + \frac{8i}{n}} \) divided by \( n \) as \( n \to \infty \) suggests an integral representing the area.
Evaluate the integral’s boundaries and function \(\sqrt{1 + \frac{8i}{n}}\) to match with the correct area in the given multiple-choice options, ensuring the limits and integrand align properly.
#### Correct Answer:
\[ \sqrt{1 + x} \text{ on } [0, 8] \]
This confirms that the integral's bounds align with the region from 0 to 8 for \( \sqrt{1 + x} \), corresponding to one of the given options.
#### Visual Representation:
No graphs or diagrams are essential for this particular solution, just the proper understanding of Riemann sums, integral boundaries, and the integration process. However, sketching the curve and the region would be beneficial for visual learners.
This educational problem aids students in grasping the concept of using limits to determine areas without directly evaluating the limit itself, reinforcing their understanding of integral calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa80fd0a7-9ae8-4bc5-b926-d69bfc10c0d7%2F11094559-6686-4e49-87de-938301624fdd%2Fpbrt80h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus: Evaluating Limits and Areas
#### Problem Statement:
Determine a region whose area is equal to the given limit. Do not evaluate the limit.
\[
\lim_{n \to \infty} \sum_{i=1}^{n} \frac{8}{n} \sqrt{1 + \frac{8i}{n}}
\]
#### Multiple Choice Options:
1. \(\sqrt{1 + x} \text{ on } [-8, 8]\)
2. \(8\sqrt{1 + x} \text{ on } [-8, 8]\)
3. \(8\sqrt{1 - x} \text{ on } [0, 8]\)
4. \(8\sqrt{1 + x} \text{ on } [0, 8]\)
5. \(\sqrt{1 + x} \text{ on } [0, 8]\)
### Solution:
Analyze the given limit and identify the region whose area it represents by interpreting the sum as a Riemann sum. The Riemann sum approximation for the area under the curve \( y = \sqrt{1 + \frac{8i}{n}} \) divided by \( n \) as \( n \to \infty \) suggests an integral representing the area.
Evaluate the integral’s boundaries and function \(\sqrt{1 + \frac{8i}{n}}\) to match with the correct area in the given multiple-choice options, ensuring the limits and integrand align properly.
#### Correct Answer:
\[ \sqrt{1 + x} \text{ on } [0, 8] \]
This confirms that the integral's bounds align with the region from 0 to 8 for \( \sqrt{1 + x} \), corresponding to one of the given options.
#### Visual Representation:
No graphs or diagrams are essential for this particular solution, just the proper understanding of Riemann sums, integral boundaries, and the integration process. However, sketching the curve and the region would be beneficial for visual learners.
This educational problem aids students in grasping the concept of using limits to determine areas without directly evaluating the limit itself, reinforcing their understanding of integral calculus.
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