Riemann sums, and what 'area under the curve' really means

The definite integral is defined as a limit of sums. We chop the interval $[a, b]$ into $n$ pieces, build a rectangle on each, add up their areas, and ask what happens as $n \to \infty$.

The setup

With a uniform partition of width $h = (b - a) / n$, a Riemann sum is

S_n = Σ  f(x_i*) · h        for i = 0 … n-1

where $x_i^*$ is some sample point inside the $i$-th subinterval. Choosing the left endpoint, right endpoint, or midpoint gives different approximations that all converge to the same integral.

Why midpoint wins

Left and right sums have error that shrinks like $O(1/n)$. The midpoint and trapezoid rules cancel the leading error term, so they shrink like $O(1/n^2)$, dramatically faster.

Don't take my word for it. Open the Riemann sums simulator, switch between methods, and watch the error column as you drag $n$. Midpoint with 12 rectangles often beats left sums with 100.