The following sum where b = and f(x) √√√√√»-()': - ( 5 )² · 5 + √/25 - (10) ².5 + ...- + n n is a right Riemann sum for the definite integral = 25- The limit of these Riemann sums as n → ∞ is [ f(x) dx 25- 5n n 2 م . مد 5 n
The following sum where b = and f(x) √√√√√»-()': - ( 5 )² · 5 + √/25 - (10) ².5 + ...- + n n is a right Riemann sum for the definite integral = 25- The limit of these Riemann sums as n → ∞ is [ f(x) dx 25- 5n n 2 م . مد 5 n
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
![**Riemann Sum and Definite Integral Evaluation**
The following sum:
\[
\sqrt{25 - \left(\frac{5}{n}\right)^2} \cdot \frac{5}{n} + \sqrt{25 - \left(\frac{10}{n}\right)^2} \cdot \frac{5}{n} + \cdots + \sqrt{25 - \left(\frac{5n}{n}\right)^2} \cdot \frac{5}{n}
\]
is a right Riemann sum for the definite integral
\[
\int_{0}^{b} f(x) \, dx
\]
where \( b = \) [ ],
and \( f(x) = \) [ ].
**The limit of these Riemann sums as \( n \to \infty \) is**
[ ].
This material is important for understanding how to approximate the area under a curve using Riemann sums, which in calculus forms the foundational concept of integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb2133c9-e1e5-4d56-9c72-044227328930%2F4f967999-281d-4a47-b7e7-6a47ea7ffd33%2Ffutfmos_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Riemann Sum and Definite Integral Evaluation**
The following sum:
\[
\sqrt{25 - \left(\frac{5}{n}\right)^2} \cdot \frac{5}{n} + \sqrt{25 - \left(\frac{10}{n}\right)^2} \cdot \frac{5}{n} + \cdots + \sqrt{25 - \left(\frac{5n}{n}\right)^2} \cdot \frac{5}{n}
\]
is a right Riemann sum for the definite integral
\[
\int_{0}^{b} f(x) \, dx
\]
where \( b = \) [ ],
and \( f(x) = \) [ ].
**The limit of these Riemann sums as \( n \to \infty \) is**
[ ].
This material is important for understanding how to approximate the area under a curve using Riemann sums, which in calculus forms the foundational concept of integration.
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