(a)
Approximate the value of integral using Trapezoidal rule with
(a)
Answer to Problem 4E
With
Explanation of Solution
Given information:
Since we are required to use the Trapezoidal rule, first we need to find the height.
Function values:
Now,
Setup the equation:
Plugin the values into the equation:
(b)
Whether the approximation is an overestimate or an under − estimate.
(b)
Answer to Problem 4E
The approximation is an overestimate.
Explanation of Solution
Given information:
First derivative of the function:
Second derivative of the function:
We know that
Second derivative gives the concavity.
If the second derivative is greater than 0,
It’s an overestimate.
If the second derivative is less than 0,
It’s an underestimate.
Note that
The value of second derivative is
Therefore,
The approximation is an overestimate.
(c)
Integral’s exact value to verify the answer.
(c)
Answer to Problem 4E
The exact value of integral is approx. 0.6931 which is not same as using the Trapezoidal rule with
Explanation of Solution
Given information:
Using FTC (Fundamental Theorem of Calculus):
Therefore,
Integral’s exact value is approx. 0.6931 which is not same as the value of integral obtained in Part (a).
Chapter 6 Solutions
Calculus: Graphical, Numerical, Algebraic
Additional Math Textbook Solutions
University Calculus: Early Transcendentals (4th Edition)
Precalculus Enhanced with Graphing Utilities (7th Edition)
Precalculus (10th Edition)
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
Thomas' Calculus: Early Transcendentals (14th Edition)
Calculus: Early Transcendentals (2nd 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