Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Please solve this question correctly.
![**Setup the Riemann Sum for the Given Integral:**
Given the integral:
\[
\int_{a}^{b} f(x) \, dx = \lim_{{n \to \infty}} \sum_{{k=1}}^{n} f(\bar{x}_k) \Delta x
\]
This is the setup for the given integral.
**Evaluate the Integral:**
\[
\int_{2}^{5} x^3 \, dx = \lim_{{n \to \infty}} \sum_{{k=1}}^{n} \left(\left(2 + \frac{3}{n}k\right)^3 \cdot \frac{3}{n}\right)
\]
The expression inside the sum represents the Riemann sum approximation for the integral of \(x^3\) from 2 to 5. Here, \(\Delta x = \frac{3}{n}\), and the sample point \(\bar{x}_k = 2 + \frac{3}{n}k\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d5d9ef2-562b-49f4-9963-a1c4e47b1118%2F47c266ab-90d6-4426-a382-124e7409bd1c%2F95ntx2ji_processed.png&w=3840&q=75)
Transcribed Image Text:**Setup the Riemann Sum for the Given Integral:**
Given the integral:
\[
\int_{a}^{b} f(x) \, dx = \lim_{{n \to \infty}} \sum_{{k=1}}^{n} f(\bar{x}_k) \Delta x
\]
This is the setup for the given integral.
**Evaluate the Integral:**
\[
\int_{2}^{5} x^3 \, dx = \lim_{{n \to \infty}} \sum_{{k=1}}^{n} \left(\left(2 + \frac{3}{n}k\right)^3 \cdot \frac{3}{n}\right)
\]
The expression inside the sum represents the Riemann sum approximation for the integral of \(x^3\) from 2 to 5. Here, \(\Delta x = \frac{3}{n}\), and the sample point \(\bar{x}_k = 2 + \frac{3}{n}k\).
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