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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The answer is on this image that I am attaching, but I'm getting confused with the trapezoidal and Simpson's rule formulas. Can you please show me how to get to this answer?
Thank You!
![**Topic: Approximating Integrals Using the Midpoint Rule**
**Objective:**
Learn how to approximate an integral using the midpoint rule with specified intervals and precision.
**Problem:**
Use the midpoint rule with \( n = 5 \) to approximate the following integral. Round to four decimal places.
\[
\int_{1}^{6} \frac{\sin x}{x} \, dx
\]
**Solution:**
The approximation calculated using the midpoint rule is: **0.4587**.
**Explanation:**
The midpoint rule is a numerical method for approximating definite integrals. It involves dividing the domain into \( n \) equal subintervals, finding the midpoint of each subinterval, then summing the function values at these midpoints, weighted by the interval width.
This method provides an estimate of the integral by approximating the area under the curve as a series of rectangles, whose heights are determined by the function values at the midpoints of each subinterval.
**Source:**
Template by Bill Arcuri, WCSD.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcfea0b1d-fbd9-4aaf-b240-b46cb35251fe%2F0d8c61ba-3466-41bc-8004-b19e8e0a655e%2Fdhcmlg5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Approximating Integrals Using the Midpoint Rule**
**Objective:**
Learn how to approximate an integral using the midpoint rule with specified intervals and precision.
**Problem:**
Use the midpoint rule with \( n = 5 \) to approximate the following integral. Round to four decimal places.
\[
\int_{1}^{6} \frac{\sin x}{x} \, dx
\]
**Solution:**
The approximation calculated using the midpoint rule is: **0.4587**.
**Explanation:**
The midpoint rule is a numerical method for approximating definite integrals. It involves dividing the domain into \( n \) equal subintervals, finding the midpoint of each subinterval, then summing the function values at these midpoints, weighted by the interval width.
This method provides an estimate of the integral by approximating the area under the curve as a series of rectangles, whose heights are determined by the function values at the midpoints of each subinterval.
**Source:**
Template by Bill Arcuri, WCSD.
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