The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.


Elementary Calculus Integral The Definite Integral Problems
 

Problems

Compute the following finite Riemann sums. If a hand calculator is available, the Riemann sums can also be computed with Dx = 1/10.

04_integration-82.gif 04_integration-83.gif
25

Let b be a positive real number and n a positive integer. Prove that if Dx = b/n,

04_integration-79.gif

Using the formula 04_integration-80.gif, prove that

04_integration-81.gif
26

Let H be a positive infinite hyperinteger and dx = b/H. Using the Transfer Principle and Problem 25, prove that04_integration-84.gifx dx = b2/2.
27

Let b be a positive real number, n a positive integer, and Dx = b/n. Using the formula

04_integration-85.gif

prove that

04_integration-86.gif
28 Use Problem 27 to show that 04_integration-87.gif x2 dx = b3/3.

Last Update: 2006-Nov-05