?Powers of x by Riemann sums Consider the integral \(I(p)=\int _0^1\ x^p\ dx\) where p
Chapter 14, Problem 334(choose chapter or problem)
Powers of x by Riemann sums Consider the integral \(I(p)=\int _0^1\ x^p\ dx\) where p is a positive integer.
a. Write the left Riemann sum for the integral with n subintervals.
b. It is a fact (proved by the 17th-century mathematicians Fermat and Pascal) that
\(\lim _{n \rightarrow \infty}\ \frac{1}{n}\ \sum _{k=0}^{n-1}\ (\frac{k}{n})^p\ =\ \frac{1}{p+1}\)
Use this fact to evaluate I(p).
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