- #1

bonfire09

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## Homework Statement

For any natural number ##p## use the preceding exercise and corollary 6.12 to prove that ##\int_0^1 x^p dx=\frac{1}{p+1}##

## Homework Equations

Preceding Exercise: Let ##p## and ##n## be natural numbers with ##n\geq 2##. Then ##\sum_{k=1}^{n-1} k^p\leq \frac{n^{p+1}}{p+1}\leq\sum_{k=1}^{n} k^p##

Corollary 6.12: Suppose the function ##f:[a,b]→\mathbb{R}## is integrable. If ##\{P_n\}## is any sequence of partitions of ##[a,b]## such that ##\lim_{n\to\infty}||P_n||=0## then ##\lim_{n\to\infty} U(f,P_n)=\lim_{n\to\infty} L(f,P_n)=\int_a^b f##.

## The Attempt at a Solution

I am really stuck on this problem for the past hour. I proved the preceding exercise already and I moved on to this one. I tried this problem using a regular partition P but that did not work. Any help would be great. Thanks