Concept explainers
(a)
To find: The expression for area under the curve
(a)
Answer to Problem 22E
The area under the curve
Explanation of Solution
Given:
The function is
The upper limit is
Formula used:
The area A of the region (S) under the graph f of a continuous function is the sum of area of the approximating rectangles is given by,
Calculation:
Find the width of the interval
Here, the upper limit is b, the lower limit is a, and the number of rectangles is n.
Substitute 1 for b and 0 for a in Equation (2).
Find the value of
Substitute 0 for a and
Use definition 2 to obtain the expression for area under the curve.
Substitute
Substitute
Therefore, the expression for the area under the curve is
(b)
To evaluate: The limit
(b)
Answer to Problem 22E
The value of the limit is
Explanation of Solution
Calculation:
The general expression of sum of cubes for the first n integers is shown below:
Rearrange the limit function as shown below.
Apply the general expression of sum of cubes in equation (1).
On further simplification,
Therefore, the value of the limit is
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