Chapter 5: Integrals - Section 5.2 - Sigma Notation and Limits of Finite Sums - Exercises 5.2 - Page 266: 45

$\dfrac{1}{2}$

Consider $f(x)=2x^3$ Here, we have $\Sigma_{i=1}^n (\dfrac{2}{n}) (k_i^3)=(\dfrac{2}{n}) \Sigma_{i=1}^n (\dfrac{i^3}{n^3}$ This implies that $(\dfrac{2}{n^4})\Sigma_{i=1}^n i^3=\dfrac{n^2+2n+1}{2n^2}$ Now, $\lim\limits_{n \to \infty}\dfrac{1+2/n+1/n^2}{2}=\dfrac{1}{2}$