Concept explainers
With a programmable calculator or computer (see the instructions for Exercise 5.1.9), compute the left and right Riemann sums for the function f(x) = x/(x + 1) on the interval [0, 2] with n = 100. Explain why these estimates show that
To Calculate: The left and right Riemann sums for the function
To Explain: the reason for the expression of estimate as follows:
Answer to Problem 14E
The value of left Riemann sums for the function
The value of right Riemann sums for the function
Explanation of Solution
The value of the function
Given:
The function as
The region lies between
Number of rectangles
Calculation:
Show the Equation of the function
Calculate the value of the function
Substitute 0 for x in Equation (1).
Substitute 1.5 for x in Equation (1).
Substitute 2 for x in Equation (1).
The calculated values of x shows that the function
The expression to find the left Riemann sum
Here, the left endpoint height of first rectangle is
Find the width
Here, the upper limit is b, the lower limit is a, and the number of rectangles is n.
Substitute 2 for b, 0 for a and 100 for n in Equation (3).
The value of left and right end points within the interval
The left endpoints are
The right endpoints are
Calculate the left Riemann sum using calculator.
Substitute 0.02 for
The value of left Riemann Sum is
The expression to find the right Riemann sum
Here, the right endpoint height of first rectangle is
Calculate the right Riemann sum using calculator.
Substitute 0.02 for
The value of right Riemann Sum is
Calculate the value of integral
Compare the left and right Riemann Sum value with value of integral
Since, the function
The value of left Riemann sum gives a lower estimate.
The value of the right Riemann sum gives a upper estimate.
Thus, expression the left and right Riemann sum as follows:
Thus, the expression for left and right Riemann sum is shown as
Chapter 5 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning