Evaluating limit from definition of integral I created a formula that uses rectangles to approximate the area under a curve. For this instance, the answer to this problem should be equal to the area under the parabola $y=x^2$ from $[0, 2]$, which is $8/3$.
$$\lim_{n\to \infty} \frac 2n \sum_{l=1}^{n-1} \frac{4l^2}{n^2}$$
I don't know how to evaluate this limit, though.  Any hints would be appreciated.

Let $S_n:=\sum_{l=1}^nl^2$. We have $S_{n}-S_{n-1}=n^2$, which is a quadratic polynomial, and $S_n$ must be a cubic polynomial.
Assume $S_n\approx an^3$ and lower powers of $n$. Then
$$S_n-S_{n-1}\approx an^3-a(n-1)^3\approx an^3-an^3+3an^2$$ and lower order terms.
Then $a=\frac13$ and your limit is $\frac83$.
