To calculate:the value of the integral
Answer to Problem 40RE
The value of the integral is
Explanation of Solution
Given information:
Formula used:
Calculation:
The given integral is
By fundamental theorem of calculus, part 1, it is given that-
If f is continuous on [ a, b ],then
Proof:
Apply the definition of the derivative directly to the function F . That is,
The expression in brackets in the last line is the average value of f from x to x + h , the mean value theorem for definite integrals. That f , being continuous, takes on its averagevalue at least once in the interval, that is,
So,
What happens to c as h goes to zero As x + h gets closer to x , it carries 0 along with it like a bead on a wire, forcing c to approach x . Since f is continuous, this means that f (c) approaches f(x):
Putting it all together,
So,
Use above proof to find
Here,
So,
Since, derivative of
Conclusion:
The value of the integral is
Chapter 6 Solutions
Calculus: Graphical, Numerical, Algebraic: Solutions Manual
Additional Math Textbook Solutions
Precalculus Enhanced with Graphing Utilities (7th Edition)
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
Precalculus (10th Edition)
Calculus: Early Transcendentals (2nd Edition)
Thomas' Calculus: Early Transcendentals (14th Edition)
Calculus & Its Applications (14th 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