On a sketch of y = In(x), represent the left-endpoint approximation with n = 2 approximating / In(x) dx. Write out the terms of the sum, ot evaluate it: her sketch, represent the right-endpoint approximation with n = 2 approximating In(x) dx. Write out the terms of the sum, but do not it: um is an overestimate? e left Riemann sum e right Riemann sum ither sum sum is an underestimate? e left Riemann sum ►
On a sketch of y = In(x), represent the left-endpoint approximation with n = 2 approximating / In(x) dx. Write out the terms of the sum, ot evaluate it: her sketch, represent the right-endpoint approximation with n = 2 approximating In(x) dx. Write out the terms of the sum, but do not it: um is an overestimate? e left Riemann sum e right Riemann sum ither sum sum is an underestimate? e left Riemann sum ►
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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![On an educational website, this text might be presented as follows:
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**Problem:**
1. On a sketch of \( y = \ln(x) \), represent the left-endpoint approximation with \( n = 2 \) approximating \( \int_1^2 \ln(x) \, dx \). Write out the terms of the sum, but do not evaluate it:
\[
\text{Sum} = \, \_\_\_ \, + \, \_\_\_
\]
2. On another sketch, represent the right-endpoint approximation with \( n = 2 \) approximating \( \int_1^2 \ln(x) \, dx \). Write out the terms of the sum, but do not evaluate it:
\[
\text{Sum} = \, \_\_\_ \, + \, \_\_\_
\]
**Questions:**
- Which sum is an overestimate?
- A. the left Riemann sum
- B. the right Riemann sum
- C. neither sum
- Which sum is an underestimate?
- A. the left Riemann sum
- B. the right Riemann sum
- C. neither sum
---
**Explanation:**
This activity involves approximating the integral of \( \ln(x) \) from 1 to 2 using Riemann sums with \( n = 2 \). The left-endpoint and right-endpoint approximations are to be visualized on sketches, and students are asked to write down the terms of each sum without calculating them.
After deriving the terms of the sums for both the left and right endpoints, students are prompted to consider which approximation method results in an overestimate or underestimate of the true value of the integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F58cffe43-d701-4c21-9740-08fe8d98ee79%2F109c84c4-7bbe-4655-87cd-307e6a39307c%2Fcqbm4l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:On an educational website, this text might be presented as follows:
---
**Problem:**
1. On a sketch of \( y = \ln(x) \), represent the left-endpoint approximation with \( n = 2 \) approximating \( \int_1^2 \ln(x) \, dx \). Write out the terms of the sum, but do not evaluate it:
\[
\text{Sum} = \, \_\_\_ \, + \, \_\_\_
\]
2. On another sketch, represent the right-endpoint approximation with \( n = 2 \) approximating \( \int_1^2 \ln(x) \, dx \). Write out the terms of the sum, but do not evaluate it:
\[
\text{Sum} = \, \_\_\_ \, + \, \_\_\_
\]
**Questions:**
- Which sum is an overestimate?
- A. the left Riemann sum
- B. the right Riemann sum
- C. neither sum
- Which sum is an underestimate?
- A. the left Riemann sum
- B. the right Riemann sum
- C. neither sum
---
**Explanation:**
This activity involves approximating the integral of \( \ln(x) \) from 1 to 2 using Riemann sums with \( n = 2 \). The left-endpoint and right-endpoint approximations are to be visualized on sketches, and students are asked to write down the terms of each sum without calculating them.
After deriving the terms of the sums for both the left and right endpoints, students are prompted to consider which approximation method results in an overestimate or underestimate of the true value of the integral.
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