Calculate the area under higher power curves

Given the inequality

$1^k + 2^k + \cdots + (n-1)^k < \frac{n^{k+1}}{k+1} < 1^k + 2^k + \cdots + n^k$

(valid for all integers $n, k \geq 1$ ), generalize the results of Exercise I.4.2 to calculate $x^n$ for any integer $n$ .

Using the same method as in the previous two exercise, here and here, we first calculate the lower and upper sums by adding up the areas of the rectangles below and above the curve, respectively.

$\begin{align*} s_n &= \frac{b}{n} \left( \frac{b}{n} \right)^k + \frac{b}{n} \left( \frac{2b}{n} \right)^k + \cdots + \frac{b}{n} \left( \frac{(n-1)b}{n} \right)^k \\[9pt] &= \left( \frac{b}{n} \right)^{k+1} (1^k + 2^k + \cdots + (n-1)^k),\\ \intertext{and,} S_n &= \frac{b}{n} \left( \frac{b}{n} \right)^k + \frac{b}{n} \left( \frac{2b}{n} \right)^k + \cdots + \frac{b}{n} \left( \frac{nb}{n} \right)^k \\[9pt] &= \left( \frac{b}{n} \right)^{k+1} (1^k + 2^k + \cdots + n^k). \end{align*}$