Difficulties with a limit while trying to calculate " integral from 0 to 1 of x² dx".

Question

This is from an exercise in Stewart, Calculus.

I managed to express the definite integral as the limit of a Riemann sum.

After having calculated

$\Delta x = \frac 1 n$

and

• right hand sample point $x_i$ = i-1/n)= i/n,

I ended with the expression

$$\lim_{n \rightarrow\infty} \sum_{i =1}^n\left (\frac{i}{n}\right)^2 \cdot \frac 1 n$$

Calculating this product is not difficult, but I cannot manage to find the limit which seems to be 0 , which is false however.

I tried to ask the question on Symbolab , but the computer does not show steps in the limit calculation, but instead uses the evaluation theorem ( using a primitive).

Answer 1

$$\lim\limits_{n\to\infty}\sum\limits_{i=1}^n \dfrac {i^2}{n^3}=\lim\limits_{n\to\infty}\dfrac{n(n+1)(2n+1)}{6n^3}.$$

Can you take it from here?