Riemann Sum

First, slice the area to n pieces.
and draw the rectangle pieces like this.

The width of the area is
1 - 0 = 1.

The area is sliced into n pieces.

So the width of each rectangle is
1/n.

The width of each piece is 1/n.

Then the x value of the right side of the kth piece is
k/n.

Draw the kth rectangle piece below.

The width is 1/n.

The height of the rectangle piece is
f(k/n) = (k/n)² = k²/n².

The width is 1/n.
The height is k²/n².

Then the area, A_k, is
A_k = (k²/n²)⋅(1/n).

There are n rectangle pieces.

So the sum of the n pieces, S_n, is
the sum of (k²/n²)⋅(1/n) as k goes from 1 to n.

Sigma Notation

The variable of the summation is k, not n.

So take the denominators, n² and n,
out from the summation.

The sum of k² is
[n(n + 1)(2n + 1)]/6.

Sum of Squares: k²

Then [n(n + 1)(2n + 1)]/6n³.

Recall that
as n → ∞,
S_n becomes the area under y = x²: S.

S_n = [n(n + 1)(2n + 1)]/6n³

So S = the limit of [n(n + 1)(2n + 1)]/6n³.

[n(n + 1)(2n + 1)]/6n³ = [2n³ + ...]/6n³

The highest order term of the numerator is 2n³.
The denominator is 6n³.

Both terms are n³.

So, as n → ∞,
the limit is 2/6.

Indeterminate Form

2/6 = 1/3

So the area of the colored region, S, is 1/3.