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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![**Estimating Integrals Using Left Riemann Sums**
**Problem Statement:**
Estimate the integral \(\int_{-3}^{3} x^3 \, dx\) using left sums and \(\Delta x = 1\).
**Explanation:**
The problem requires estimating the value of the integral \(\int_{-3}^{3} x^3 \, dx\) using the left Riemann sum method with a partition width (\(\Delta x\)) of 1.
### Riemann Sum Method:
A Riemann sum is an approximation of an integral by a finite sum, typically used when calculating the definite integral of a function over an interval. In this case, we're using left sums, which means the height of each rectangle is determined by the function value at the left endpoint of each subinterval.
### Steps to Solve:
1. **Determine the subintervals**: The interval \([-3, 3]\) is divided into subintervals of width \(\Delta x = 1\).
2. **Calculate the left endpoints**: The left endpoints for each subinterval are \(-3, -2, -1, 0, 1, 2\).
3. **Evaluate the function at each endpoint**: Calculate \(f(x) = x^3\) for each left endpoint.
4. **Calculate the sum**: Multiply each function value by \(\Delta x\) and then sum all these products to approximate the integral.
This method is especially useful for educational purposes as it provides a practical approach to understanding integrals through approximation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6a025add-db4b-4f7d-8d83-fda6c2b9cf69%2F6d838278-94d4-43d8-9dcf-242cccb7d2cf%2Fx5561jr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Estimating Integrals Using Left Riemann Sums**
**Problem Statement:**
Estimate the integral \(\int_{-3}^{3} x^3 \, dx\) using left sums and \(\Delta x = 1\).
**Explanation:**
The problem requires estimating the value of the integral \(\int_{-3}^{3} x^3 \, dx\) using the left Riemann sum method with a partition width (\(\Delta x\)) of 1.
### Riemann Sum Method:
A Riemann sum is an approximation of an integral by a finite sum, typically used when calculating the definite integral of a function over an interval. In this case, we're using left sums, which means the height of each rectangle is determined by the function value at the left endpoint of each subinterval.
### Steps to Solve:
1. **Determine the subintervals**: The interval \([-3, 3]\) is divided into subintervals of width \(\Delta x = 1\).
2. **Calculate the left endpoints**: The left endpoints for each subinterval are \(-3, -2, -1, 0, 1, 2\).
3. **Evaluate the function at each endpoint**: Calculate \(f(x) = x^3\) for each left endpoint.
4. **Calculate the sum**: Multiply each function value by \(\Delta x\) and then sum all these products to approximate the integral.
This method is especially useful for educational purposes as it provides a practical approach to understanding integrals through approximation.
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