Stacking Rectangles

Using LRAM with $n = 11$ , we partition the interval [0, 1] into 11 equal subintervals:

$\left[0, \dfrac{1}{11}\right] , \left[\dfrac{1}{11}, \dfrac{2}{11}\right], \ldots , \left[\dfrac{9}{11}, \dfrac{10}{11}\right], \left[\dfrac{10}{11}, 1\right]$

Using only the left endpoints and plugging them into $x^3$ gives $\sum_{n=0}^{10} \frac{n^3}{11^3} = \frac{1^3 + 2^3 + \ldots + 9^3 + 10^3}{11^3} = \frac{\left(\frac{(10)(11)}{2}\right)^2}{11^3} = \frac{25}{121} \approx 0.2066$