The Riemann Integral: Partitions, Darboux Sums and Integrability Criteria

The Riemann integral ∫_a^b f(x) dx is the limit of sums f(x₁)Δx₁ + … + f(x_n)Δx_n over finer and finer partitions of [a, b]; a bounded function is integrable exactly when its upper and lower Darboux sums can be made arbitrarily close. In brief: for f(x) = x² on [0, 1], the right-endpoint sum with n subintervals is (n + 1)(2n + 1)/(6n²), which tends to 1/3. Continuous functions and monotone functions are integrable, and a function with a jump at finitely many points is integrable as well.

The integral gives a precise meaning to “the area under a curve” and to total accumulation. Its definition by sums is also the starting point for numerical integration, and it determines which functions can be integrated at all. This guide defines partitions, Riemann sums and Darboux sums, states the integrability criterion, gives the main properties of the integral and the mean value theorem for integrals, and shows computations with a table of approximations for x².

What are partitions and Riemann sums?

A partition P of [a, b] is a finite set of points a = x₀ < x₁ < … < x_n = b. It divides the interval into subintervals [x_i−1, x_i] of length Δx_i = x_i − x_i−1. The norm ‖P‖ is the length of the longest subinterval. A Riemann sum is S(P, ξ) = Σ f(ξ_i)Δx_i, where the sample point ξ_i is any point of the i-th subinterval (left endpoint, right endpoint, midpoint, …). Each term is the area of a rectangle with base Δx_i and height f(ξ_i).

The function f is Riemann integrable on [a, b] with integral I if for every ε > 0 there is δ > 0 such that |S(P, ξ) − I| < ε for every partition with ‖P‖ < δ and every choice of sample points. We write I = ∫_a^b f(x) dx. For f ≥ 0, the integral is the area between the graph and the x-axis. The antiderivative is a different concept, connected with the definite integral by the fundamental theorem of calculus.

What are Darboux sums?

For a bounded function f, let m_i = inf f and M_i = sup f on the i-th subinterval. The lower Darboux sum is L(P) = Σ m_iΔx_i and the upper Darboux sum is U(P) = Σ M_iΔx_i. Every Riemann sum lies between them: L(P) ≤ S(P, ξ) ≤ U(P). Refining the partition (adding points) increases L and decreases U. So every lower sum is at most every upper sum, and the lower integral sup_P L(P) is at most the upper integral inf_P U(P).

When is a function integrable?

Darboux criterion: f is integrable if and only if the lower and the upper integrals are equal; equivalently, for every ε > 0 there is a partition P with U(P) − L(P) < ε. The common value is the integral.

• Continuous functions on [a, b] are integrable (they are uniformly continuous, so M_i − m_i is small on small intervals).
• Monotone functions are integrable: U(P) − L(P) ≤ (f(b) − f(a))‖P‖ for an increasing f.
• Bounded functions with finitely many discontinuities (jumps, holes) are integrable. More generally, a bounded function is integrable if and only if its set of discontinuities has Lebesgue measure zero (Lebesgue’s criterion).
• Unbounded functions are not Riemann integrable.
• The Dirichlet function D(x) = 1 for rational x and 0 for irrational x is not integrable on [0, 1]: every subinterval contains both kinds of numbers, so L(P) = 0 and U(P) = 1 for every partition.

What are the properties of the integral?

• Linearity: ∫ (αf + βg) = α ∫ f + β ∫ g.
• Additivity: ∫_a^b f = ∫_a^c f + ∫_c^b f for a < c < b.
• Orientation: ∫_b^a f = −∫_a^b f, and ∫_a^a f = 0.
• Monotonicity: if f ≤ g on [a, b], then ∫ f ≤ ∫ g. In particular, if m ≤ f ≤ M, then m(b − a) ≤ ∫ f ≤ M(b − a).
• Absolute value: |∫ f| ≤ ∫ |f|.
• Mean value theorem for integrals: if f is continuous on [a, b], there is c in [a, b] with ∫_a^b f = f(c)(b − a). The number f(c) = (1/(b − a)) ∫ f is the average value of f.
• Symmetry: for an odd function ∫_−a^a f = 0, and for an even function ∫_−a^a f = 2 ∫₀^a f.

Worked examples

Example 1: ∫₀¹ x² dx from the definition

Use n equal subintervals, Δx = 1/n, and right endpoints x_k = k/n. The sum is Σ (k/n)²(1/n) = (1/n³) Σ k² = (1/n³) · n(n + 1)(2n + 1)/6 = (n + 1)(2n + 1)/(6n²) = (2n² + 3n + 1)/(6n²) → 1/3.

Example 2: a linear function

∫₀² (x + 1) dx is the area of a trapezoid with parallel sides 1 and 3 and height 2: (1 + 3)/2 · 2 = 4. By Riemann sums with n subintervals and right endpoints: Σ (2k/n + 1)(2/n) = (4/n²)(n(n + 1)/2) + 2 = 2(n + 1)/n + 2 → 4 ✓.

Example 3: a geometric area

∫₋₁¹ √(1 − x²) dx is the area of a half of the unit disc: π/2 ≈ 1.571. And ∫₀³ |x − 1| dx = (1/2)(1)(1) + (1/2)(2)(2) = 2.5, as the sum of two triangles.

Example 4: the Darboux criterion

For increasing f(x) = x² on [0, 1] with n equal parts, L(P) is the left sum (n − 1)(2n − 1)/(6n²) and U(P) is the right sum (n + 1)(2n + 1)/(6n²). U − L = [(2n² + 3n + 1) − (2n² − 3n + 1)]/(6n²) = 1/n, which can be made smaller than any ε. So x² is integrable, and the integral is the common limit 1/3.

Example 5: average value

The average of f(x) = x² on [0, 3]: (1/3) ∫₀³ x² dx = (1/3)(9) = 3. By the mean value theorem for integrals, f(c) = c² = 3 for c = √3 ≈ 1.732 in [0, 3].

Example 6: a bound without computing

For f(x) = 1/(1 + x²) on [0, 1], 1/2 ≤ f ≤ 1, so 1/2 ≤ ∫₀¹ f ≤ 1. The actual value is π/4 ≈ 0.785.

Example 7: a function with one jump

f(x) = 0 for x < 1 and f(x) = 2 for x ≥ 1 on [0, 3]. It is integrable (one discontinuity), and ∫₀³ f = 0 · 1 + 2 · 2 = 4.

Table of approximations of ∫ x² dx

The integral 1/3 ≈ 0.333333 always lies between the lower and upper sums.

Where does the definition come from?

Archimedes computed areas by exhaustion, an early form of the limit of sums. Newton and Leibniz defined the integral as the inverse of the derivative. Augustin-Louis Cauchy defined the integral of a continuous function as the limit of sums in 1823. Bernhard Riemann extended the definition to a wider class of functions in his 1854 habilitation thesis, and Gaston Darboux in 1875 introduced the upper and lower sums that simplify the theory. Henri Lebesgue’s theory of integration (1902) is more general and removes the restrictions of the Riemann integral.

Where is it used?

Riemann sums are the basis of numerical integration (the rectangle, trapezoid and Simpson rules) used in engineering and in computer programs to compute areas, work and probabilities. In physics, total distance is the integral of speed, total charge the integral of current. In statistics, the probability that a continuous random variable lies in an interval is the integral of its density.

Which mistakes are common?

Confusing the definite integral with the antiderivative

The definite integral is a number, a limit of sums. The indefinite integral is a family of functions. The fundamental theorem connects them.

Assuming that the integral equals the area

For functions that change sign, the integral is the signed area: regions below the x-axis count negatively. The integral of sin x over [0, 2π] is 0, but the area is 4.

Using unequal rectangles without the proper widths

In a Riemann sum, each height is multiplied by its own width Δx_i.

Assuming that any function can be integrated

The Dirichlet function is not Riemann integrable, and unbounded functions need improper integrals.