Riemann Sums – Styles, Formulas, and Conceptual Insights

Purpose: first-principles method for finding the exact value of a definite integral (area under y=f(x) on a closed interval [a,b]) by adding up areas of thin rectangles and letting their width go to zero.
A Riemann sum is the finite summation that precedes the limiting process; in the limit it becomes the integral.
Conceptual picture:
- Partition [a,b] into n sub-intervals.
- On each sub-interval, erect a rectangle whose height is sampled from the function.
- Add all rectangular areas: width \times height.

Notation for the Riemann sum:
Ingredients:
- \Delta x = \dfrac{b-a}{n} (common width of each rectangle when the partition is uniform).

x_i^* specifies where in the i-th sub-interval the height is read from the curve.
Different rules for picking x_i^* generate different *styles* of Riemann sums; each style yields a distinct finite approximation even though they all converge to the same integral (for integrable functions).

Uses the exact center, often improves accuracy for the same n because error terms cancel symmetrically.

Upper sum: choose x_i^* in each sub-interval where f attains its *maximum* value.
- Guarantees every rectangle overestimates area: \text{Upper Sum} \ge \int_a^b f(x)\,dx.
Lower sum: choose x_i^* where f attains its *minimum* value.
- Guarantees under-estimation: \text{Lower Sum} \le \int_a^b f(x)\,dx.
Useful for bounding the true integral from above and below; difference goes to zero if and only if f is Riemann-integrable.
No simple closed-form formula for x_i^* because it depends on the function’s local extrema.

Rule: select each x_i^* randomly (uniformly) within its sub-interval.
Motivation / advantages:
- Avoids systematic over- or under-estimation biases that might occur with highly structured sampling rules.
- Mirrors probabilistic methods; heavily used in numerical integration of high-dimensional problems.
Heights = independent random variables f(x_i^*); expectation converges to the integral by the Law of Large Numbers.

If the limit of any (and thus every) style of Riemann sum exists as n \to \infty, the function is Riemann-integrable on [a,b] and
\inta^b f(x)\,dx = \lim{n\to\infty}\sum{i=1}^{n} f\bigl(xi^*\bigr)\,\Delta x.
Different finite sums yield different numerical values for small n, but all share the same limit when n becomes large.
Upper vs. lower sums squeeze the integral from above and below; equality of their limits is a standard integrability criterion.

All Riemann sum styles rely on the same geometric idea: rectangle widths equal, heights pegged to f(x).
Right-endpoint rule is often favored for simplicity and because it lines up nicely when adding new sub-intervals.
Monte Carlo sampling is “bias-free” precisely because it lacks a deterministic pattern—helpful in avoiding pathological coincidences.
Over/under-estimation properties make upper & lower sums valuable when one needs guaranteed bounds (e.g.
in proofs or error estimates).
Even though stylistic choices change intermediate approximations, the integral is unique; “any of them will work.”