Evaluate [ f(x) dx = lim f(F₁) Ar. n→∞ k=1 (a² +5a) da using the infinite Riemann sum
Evaluate [ f(x) dx = lim f(F₁) Ar. n→∞ k=1 (a² +5a) da using the infinite 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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![**Calculus Problem: Evaluating a Definite Integral using Riemann Sums**
Evaluate the integral:
\[
\int_{1}^{6} (x^2 + 5x) \, dx
\]
using the infinite Riemann sum, expressed as:
\[
\int_{a}^{b} f(x) \, dx = \lim_{n \to \infty} \sum_{k=1}^{n} f(\bar{x}_k) \Delta x
\]
**Explanation:**
The problem involves evaluating a definite integral of the polynomial function \(x^2 + 5x\) over the interval from 1 to 6. This can be approached through the concept of Riemann sums, where the integral is viewed as the limit of the sum of areas of rectangles as their number approaches infinity (and their width approaches zero). Here, \(\Delta x\) represents the width of each rectangle and \(\bar{x}_k\) is a sample point in each subinterval \([x_{k-1}, x_k]\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d5d9ef2-562b-49f4-9963-a1c4e47b1118%2Fc5620b99-6bc9-4ba1-9aee-28ab10067b18%2F5etzp4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Calculus Problem: Evaluating a Definite Integral using Riemann Sums**
Evaluate the integral:
\[
\int_{1}^{6} (x^2 + 5x) \, dx
\]
using the infinite Riemann sum, expressed as:
\[
\int_{a}^{b} f(x) \, dx = \lim_{n \to \infty} \sum_{k=1}^{n} f(\bar{x}_k) \Delta x
\]
**Explanation:**
The problem involves evaluating a definite integral of the polynomial function \(x^2 + 5x\) over the interval from 1 to 6. This can be approached through the concept of Riemann sums, where the integral is viewed as the limit of the sum of areas of rectangles as their number approaches infinity (and their width approaches zero). Here, \(\Delta x\) represents the width of each rectangle and \(\bar{x}_k\) is a sample point in each subinterval \([x_{k-1}, x_k]\).
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