1 Definition and geometric intuition

1.1 Partitioning an interval

A Riemann sum starts by choosing a finite interval \([a,b]\) and splitting it into smaller pieces. This is done by selecting points \[ a=x_0<x_1<\cdots<x_n=b, \] which determine subintervals \([x_{i-1},x_i]\). The lengths \(\Delta x_i=x_i-x_{i-1}\) control how fine the approximation is: larger subintervals produce a rough picture, while smaller ones tend to better capture the behavior of the function.

1.2 Sample points and rectangles

To form the sum, one chooses a sample point \(t_i\) inside each subinterval \([x_{i-1},x_i]\). The function value \(f(t_i)\) is used as the height of a rectangle over the width \(\Delta x_i\). The sum of the rectangle areas approximates the total area under the curve \(y=f(x)\) between \(a\) and \(b\).

1.3 The basic Riemann sum formula

With sample points \(t_i\in [x_{i-1},x_i]\), the associated Riemann sum is \[ \sum_{i=1}^{n} f(t_i)\,\Delta x_i. \] This quantity depends on both the partition and the choice of sample points. By varying these choices in a controlled way, one can obtain approximations that approach the definite integral when the function is integrable.

1.4 Left, right, and midpoint interpretations

Common special choices of sample points include:

  • Left-endpoint sums: \(t_i=x_{i-1}\), using the value at the left edge of each subinterval.
  • Right-endpoint sums: \(t_i=x_i\), using the value at the right edge.
  • Midpoint sums: \(t_i=\frac{x_{i-1}+x_i}{2}\), using the midpoint of each subinterval.

These variants can yield different approximations for finite partitions, but under suitable conditions they all converge to the same limit as the partition becomes finer.

2 Connection to the Riemann integral

2.1 Definite integral as a limit of sums

The Riemann integral can be understood as the value that Riemann sums approach when partitions are refined and the subinterval widths shrink.

2.1.1 Refining partitions

Refinement means using partitions with more (and thus smaller) subintervals. A refinement process typically produces a sequence of partitions whose maximum subinterval length decreases toward zero. This ensures that each rectangle becomes a better local approximation.

2.1.1.1 Mesh size and its role in convergence

A key quantity is the mesh size, defined by \[

\|\mathcal{P}\|=\max_{1\le i\le n}(x_i-x_{i-1})

\]

for a partition \(\mathcal{P}=\{x_0,\dots,x_n\}\). As \(\|\mathcal{P}\|\to 0\), the rectangles align more closely with the curve, enabling convergence when the function behaves appropriately.

2.1.2 Convergence of Riemann sums

A function \(f\) on \([a,b]\) is Riemann integrable if there exists a number \(I\) such that for every sequence of partitions whose mesh tends to zero (with suitable choices of sample points), the corresponding Riemann sums converge to \(I\). Intuitively, the integral is the consistent “limiting area” despite local fluctuations.

2.2 Equivalent formulations

Several reformulations capture integrability using upper and lower approximations or criteria involving oscillation.

2.2.1 Using upper and lower sums

For each partition, one can define:

  • the upper sum as a weighted sum using the supremum of \(f\) on each subinterval, and
  • the lower sum using the infimum of \(f\) on each subinterval.

If, as partitions are refined, the upper and lower sums approach the same limit, that common limit is the integral. This approach emphasizes bounding rather than tracking specific rectangle heights.

2.2.2 Darboux criterion and integrability

The Darboux criterion states that \(f\) is Riemann integrable if and only if the gap between the upper and lower sums can be made arbitrarily small by choosing a sufficiently fine partition. Formally, the integral exists exactly when the least upper bound of all lower sums equals the greatest lower bound of all upper sums.

3 Choosing points and types of Riemann sums

3.1 Fixed endpoint rules

Left- and right-endpoint sums correspond to deterministic sampling. For monotone functions, these can provide guaranteed one-sided bounds: left sums lie on one side of the true area and right sums on the other, with both tightening as the partition is refined. For general functions, they may still converge, but their approximation quality depends on how \(f\) oscillates within each subinterval.

3.2 Arbitrary point (general) Riemann sums

A general Riemann sum allows \(t_i\) to be any choice in each subinterval. This flexibility is central to the Riemann integral’s definition: integrability requires that no matter how the sample points are selected (as long as they remain within their subintervals), the resulting sums converge to the same value as the mesh size shrinks.

3.3 Midpoint Riemann sums

Midpoint sums often provide better approximations in practice because, when \(f\) is reasonably smooth, sampling in the center of each subinterval tends to reduce systematic bias from local curvature. While this is an intuitive advantage, convergence to the same integral value still follows under the integrability framework.

3.4 Step functions and interpretation

An instructive connection is that Riemann sums can be interpreted as integrals of simple step functions built from the partition. Given a partition and chosen sample heights \(f(t_i)\), one forms a step function that is constant on each subinterval and equals \(f(t_i)\) there. Then the Riemann sum is exactly the area (integral) of that step function over \([a,b]\), linking approximation by step functions to the limiting process that defines the integral.

4 Conditions for integrability

4.1 Riemann integrable functions

Riemann integrability is tied to how “wildly” a function can oscillate.

4.1.1 Continuity and discontinuities

If \(f\) is continuous on \([a,b]\), it is automatically Riemann integrable. Continuity prevents uncontrolled oscillation on small scales, ensuring that upper and lower sums become close as partitions refine. Discontinuities are not automatically disqualifying, but they must be sufficiently limited in structure.

4.1.2 Measure-zero discontinuities (criterion overview)

A common characterization is that a bounded function is Riemann integrable if its set of discontinuities is sufficiently small in the sense of having measure zero (for instance, containing no “substantial length” intervals). In many textbook treatments, this is summarized by the idea that discontinuities may occur, but they cannot occupy a significant portion of the interval in terms of length. This aligns with the Darboux perspective: the upper–lower gap can shrink to nothing if the discontinuities are confined.

4.2 Implications of boundedness

Riemann integration typically requires the function to be bounded on \([a,b]\). If \(f\) becomes unbounded, rectangle heights can grow without limit on some subintervals, preventing upper sums from stabilizing. Thus boundedness is a practical prerequisite for the upper/lower-sum gap to become arbitrarily small.

4.3 Examples of integrable vs non-integrable functions

  • Integrable example: A continuous function like \(f(x)=x^2\) on \([0,1]\) is integrable. Any reasonable Riemann sum converges to the same value.
  • Integrable with discontinuities: A function with finitely many jump discontinuities is still Riemann integrable because the set of discontinuities has negligible “size.”
  • Non-integrable example: Functions that oscillate infinitely often in a way that prevents the upper and lower sums from collapsing—often due to discontinuities spread throughout the interval—can fail Riemann integrability. In such cases, no matter how fine the partition, the supremum-infimum gap persists.

5 Properties and algebraic rules

5.1 Linearity of the integral via Riemann sums

Riemann sums satisfy algebraic identities that pass to the limit. If \(f\) and \(g\) are integrable and \(c\) is a constant, then the integral respects linear combinations: \[ \int_a^b (cf+g)=c\int_a^b f+\int_a^b g. \] Conceptually, rectangle areas scale and add exactly, and taking the limit preserves those relationships.

5.2 Additivity over subintervals

The integral is additive across adjacent intervals: if \(a<c<b\) and \(f\) is integrable on both \([a,c]\) and \([c,b]\), then it is integrable on \([a,b]\), and \[ \int_a^b f=\int_a^c f+\int_c^b f. \] In Riemann-sum terms, partitions can be refined so that \(c\) is a partition point, splitting the sum into two parts.

5.3 Monotonicity and comparison principles

If \(f\le g\) pointwise on \([a,b]\), then \[ \int_a^b f\le \int_a^b g. \] This follows from rectangle comparisons: each rectangle height for \(f\) is at most that of \(g\), so the associated sums maintain the inequality for every partition, and the limit preserves it. Similar comparison arguments lead to standard bounds for integrals.

5.4 Bounds using sum inequalities

Upper and lower bounds can be obtained directly from Riemann sums. For any partition, the lower sum does not exceed the integral and the upper sum does not fall below it. Hence, by computing tight bounds from a chosen partition (or by refining it), one can bracket the integral value without calculating it exactly.

6 Error and approximation

6.1 Approximating integrals numerically

In numerical settings, integrals are approximated by Riemann sums with finitely many subintervals. Depending on the chosen sampling rule (left, right, midpoint, or general), one obtains different estimators for the same underlying integral. Refining the partition typically improves accuracy, though computational cost increases.

6.2 Refinement and convergence rates (intuitive)

Convergence refers to how quickly Riemann sums approach the integral as the mesh shrinks. The intuitive idea is that smaller subintervals reduce the discrepancy between the function and the rectangle model. How fast this happens depends on the function’s regularity: smoother functions tend to yield faster convergence, while highly oscillatory functions require much finer partitions to achieve comparable accuracy.

6.3 Smoothness assumptions and improved accuracy (overview)

When \(f\) has additional structure—such as differentiability or bounds on its rate of change—more informative error estimates become available. For instance, midpoint sampling often benefits from curvature information, and trapezoidal-type constructions (closely related to endpoint sampling) can be more accurate for functions with controlled second derivatives. These refinements typically arise from comparing the rectangle approximation to local Taylor expansions.

6.4 Relationship to numerical quadrature concepts

Riemann sums are the conceptual foundation for broader numerical quadrature methods. While quadrature rules may combine endpoint values, midpoints, or weighted samples to achieve higher-order accuracy, the core principle remains the same: approximate an integral by a weighted sum of function evaluations, then analyze how the approximation behaves as the discretization is refined.

7.1 Riemann–Stieltjes integral (conceptual connection)

The Riemann–Stieltjes integral generalizes the Riemann integral by replacing the role of \(dx\) with increments of a function \(g\). Instead of rectangle widths \(\Delta x_i\), one uses \(\Delta g_i=g(x_i)-g(x_{i-1})\). This framework is useful when the “weight” across the interval is naturally described by another function rather than by ordinary length.

7.2 Upper/lower sums vs general sums

Upper and lower sums provide a robust way to define and test integrability, especially when sample-point choices might be difficult to control. General Riemann sums show that, once integrability holds, even arbitrary sampling within subintervals yields the same limiting value. Together, these approaches connect bounding logic with pointwise sampling.

Step functions are integrable in a straightforward way, and their integrals are computed exactly as finite sums. Riemann sums can be seen as integrals of step functions constructed from partitions. The Riemann integral then appears as the limit of these exact step-function integrals as the partition becomes finer.

7.4 Historical development and terminology

The terminology “Riemann sum” reflects its role in motivating and formalizing the Riemann integral. Historically, the development of rigorous integration required moving beyond intuitive area arguments to definitions that handle discontinuities and limits. The use of rectangles and partitions provided both an accessible geometric picture and a precise algebraic method, enabling the eventual establishment of integrability criteria and the formal study of convergence.