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


Problems

Compute the following finite Riemann sums. If a hand calculator is available, the Riemann sums can also be computed with Δx = 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 Δx = 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 Δx = 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: 2010-11-26