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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Question
find the riemann sum
![**Problem Statement:**
If \( f(x) = 2x^2 - 8 \), \( 0 \leq x \leq 3 \), find the Riemann sum with \( n = 6 \), taking the sample points to be midpoints.
**Question:**
What does the Riemann sum represent? Illustrate with a diagram.
**Diagram Explanation:**
A typical illustration for this problem would involve a graph of the function \( f(x) = 2x^2 - 8 \) over the interval [0, 3]. The interval is divided into 6 sub-intervals of equal width. Each sub-interval's midpoint is used as a sample point to evaluate the function. Rectangles are drawn such that their height corresponds to the function value at these midpoints, and their width is the width of each sub-interval. The sum of the areas of these rectangles approximates the area under the curve, which is the Riemann sum.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa699feee-67df-45a7-b712-d93645cac786%2F4ab176d7-4589-438b-a68d-45b021108c8a%2F00zeur_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
If \( f(x) = 2x^2 - 8 \), \( 0 \leq x \leq 3 \), find the Riemann sum with \( n = 6 \), taking the sample points to be midpoints.
**Question:**
What does the Riemann sum represent? Illustrate with a diagram.
**Diagram Explanation:**
A typical illustration for this problem would involve a graph of the function \( f(x) = 2x^2 - 8 \) over the interval [0, 3]. The interval is divided into 6 sub-intervals of equal width. Each sub-interval's midpoint is used as a sample point to evaluate the function. Rectangles are drawn such that their height corresponds to the function value at these midpoints, and their width is the width of each sub-interval. The sum of the areas of these rectangles approximates the area under the curve, which is the Riemann sum.
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