A definite integral is the limiting signed accumulation of small contributions.
Key knowledge
For a continuous real function on [a,b], the definite integral is the limit of sums f(xᵢ*)Δx over successively finer equal subdivisions. It measures signed accumulation; negative contributions subtract rather than become positive area.
Values and equations
∫ₐᵇ f(x) dx = limₙ→∞ Σᵢ₌₁ⁿ f(xᵢ*)Δx
f
Continuous real-valued function on [a,b]
a, b
Finite integration bounds with a < b
n
Number of equal subintervals
xᵢ*
Sample point in subinterval i
Δx
Subinterval width (b−a)/n
Conditions and scope
Here a < b and f is continuous, a sufficient integrability condition.
Choose xᵢ* within each subinterval and Δx=(b−a)/n.
This entry does not cover improper integrals or every integrable discontinuous function.
Sources · 3
References
Sources retain their recorded scope. A source link alone does not establish verification of every statement.
2026-10-10 · Checked subdivision and sign conventions; continuity stated as sufficient, not necessary.
Foundational review
2026-10-10 · Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication.
Sources from other places
Scope or convention differs
Sampled subdivisions approach a Riemann integral in the positive-area construction.
The inspected construction is narrower than the repository signed continuous-function statement.
Universidade de São Paulo, Instituto de Matemática e Estatística
For a continuous real function on [a,b], the definite integral is the limit of sums f(xᵢ*)Δx over successively finer equal subdivisions. It measures signed accumulation; negative contributions subtract rather than become positive area.
For bounded real f on [a,b], lower and upper Darboux sums use each subinterval's infimum and supremum. The Riemann integral exists when the supremum of lower sums equals the infimum of upper sums; equivalently, for every ε>0 some partition has U(f,P)−L(f,P)<ε.
On n equal subintervals with h=(b−a)/n, the trapezoidal estimate is T=h[f(a)/2+Σᵢ₌₁ⁿ⁻¹f(a+ih)+f(b)/2]. For C² f on the finite interval its error is O(h²) as h→0.
A regular C¹ plane or space curve r(t), a≤t≤b, has length ∫ₐᵇ‖r′(t)‖dt. For a C² regular space curve its curvature is κ=‖r′×r″‖/‖r′‖³, equivalently ‖dT/ds‖ with unit tangent T and arc length s.
For continuous real 2π-periodic boundary data g, u(r,θ)=(2π)⁻¹∫₋π^π[(1−r²)/(1−2r cos(θ−α)+r²)]g(α)dα is harmonic on 0≤r<1 and extends continuously to u(1,θ)=g(θ). It is the unique such harmonic solution on the closed unit disk.
With even n≥2 and h=(b−a)/n, Sₙ=(h/3)[f(a)+f(b)+4Σodd i f(a+ih)+2Σeven i f(a+ih)], summing interior indices. For f∈C⁴[a,b], |∫ₐᵇf−Sₙ|≤(b−a)h⁴ max|f⁽⁴⁾|/180. It is exact for polynomials of degree at most three.
For a complex 2π-periodic Riemann-integrable f, cₙ=(2π)⁻¹∫₋π^πf(x)e⁻ⁱⁿˣdx and s_N=Σ|n|≤N cₙeⁱⁿˣ satisfy ∫|f−s_N|²→0. Parseval gives (2π)⁻¹∫₋π^π|f|²=Σₙ∈Z|cₙ|². This is mean-square convergence, which does not assert pointwise convergence everywhere.
For solar flux S perpendicular to the rays at the current orbital distance,latitudeφ and declinationδ give cosZ=sinφsinδ+cosφcosδcosh. Integrating positive illumination over a rotation gives meanQ=(S/π)[h₀sinφsinδ+cosφcosδsinh₀]. For ordinary sunrise/sunset h₀=arccos(−tanφtanδ); polar-day/night limits use h₀=π or0 with the appropriate geometry.
Rotation-average energy is the stated finite integral.
For a planar shear-fault source with uniform shear modulus μ and consistently directed slip magnitude D, scalar moment is M₀=μAD̄, equivalently μ∫_A D dA. Moment magnitude uses Mw=(2/3)[log₁₀(M₀/(1N m))−9.1]. Source moment describes fault-slip size and is distinct from radiated energy or intensity at a site.
The area-weighted source sum is a bounded integral.
In a high-frequency ray model, travel time along a pathγ is t=∫_γ ds/v. Across a stationary planar interface, matching phase along the interface gives sinθ₁/v₁=sinθ₂/v₂ for propagating branches, with angles from the normal. For v₂>v₁ the transmitted grazing angle occurs at sinθ_c=v₁/v₂. Reflection and converted elastic branches require their own speeds and boundary conditions.
For an infinite line parallel to y at horizontal coordinate x₀ and depth b>0 with signed excess mass per lengthm_l, its downward surface anomaly is Δg=2Gm_lb/[(x−x₀)²+b²]. Integrating lines from x₁ to x₂ gives a thin strip model Δg=2GΔρH[arctan((x₂−x)/b)−arctan((x₁−x)/b)]. Greater burial broadens a fixed line-mass anomaly and lowers its peak magnitude.
With depthz positive downward, constant conductivityk and uniform volumetric heat productionH, a stationary conductive slab obeys kT_zz+H=0. If T(0)=T_s and outward/upward surface heat flux is q_s=kT_z(0), then T(z)=T_s+(q_s/k)z−Hz²/(2k). At H=0 the gradient is constant; internal production makes upward flux decrease with depth in this model.
Integrating a constant-coefficient second-order equation fixes the geotherm.