An integral is a mathematical object that generalizes the concept of area under a curve.The definite integral ∫ₐᵇ f(x)dx gives a number — the area of a curvilinear trapezoid bounded by the graph of f(x), the x-axis, and verticals x=a and x=b. It is defined as the limit of Riemann sums: Σ f(ξᵢ)Δxᵢ as the maximum length of partition intervals tends to zero. The indefinite integral is a family of antiderivatives, i.e., functions F such that F'=f. The connection is given by the Newton-Leibniz formula: ∫ₐᵇ f = F(b)−F(a).
Eudoxus (c. 370 BC) and Archimedes (287–212 BC) used the method of exhaustion, the ancient analog of integration. In the 17th century, independent works of Newton and Leibniz formalized the integral as the inverse of the differential. Riemann in 1854 gave the first formal definition, and in the early 20th century, Henri Lebesgue constructed an integral based on measure, solving fundamental problems of interchange of limits and integral.
The fundamental theorem of calculus: if f is continuous on [a,b] and F is any antiderivative, then ∫ₐᵇ f(x)dx = F(b)−F(a). Computation reduces to finding an antiderivative. Techniques: substitution, integration by parts (derivative of a product), partial fractions. For complex integrands, numerical methods (trapezoidal, Simpson's) or Fourier/Laplace transforms are used.
Calculation of distance traveled from a speed graph, area of a complex shape, prediction of accumulated radiation dose in materials — all through the integral.
The integral sign ∫ is a stylized letter ſ (long s), from Latin summa, proposed by Leibniz. Unlike the derivative, the integral smooths a function: even a highly jagged curve can have a smooth integral function.
| i | variable | inclination angle of the orbital plane — inclination angle radian |
| b | variable | minor axis of the projected orbit — minor axis metre |
| a | variable | major axis of the projected orbit — major axis metre |
| cos | operator | cosine function — cosine function |
| N | variable | Number of tiles needed — count |
| A | variable | Total area to cover — area square metre |
| a | variable | Area of one tile — area square metre |
| ⌈ ⌉ | operator | Rounds up to the nearest integer — ceiling function |
| T | variable | tunneling probability — transmission probability |
| z1 | variable | left turning point of the barrier — position metre |
| z2 | variable | right turning point of the barrier — position metre |
| m | variable | mass of the tunneling particle — mass kilogram |
| V | variable | potential energy as a function of position — potential energy joule |
| E | variable | energy of the particle — energy joule |
| ħ | constant | fundamental quantum of action divided by 2π — reduced planck constant = 1.054571817e-34 joule second |
| ∫ | operator | sums the integrand over the interval — integral |
| √ | operator | takes the non-negative root — square root |
| exp | operator | raises e to the given power — exponential |
| T_{\mu\nu} | variable | stress-energy tensor — stress energy tensor joule per metre cubed |
| k^\mu | variable | null vector — null vector |
| t | variable | time coordinate — time second |
| f(t) | variable | sampling function — sampling function |
| C | variable | dimensionless constant — dimensionless constant |
| t_0 | variable | characteristic timescale — time second |
| \int_{-\infty}^{\inf | operator | integration over all time — integral |
| DM | variable | Dispersion measure — dispersion measure parsec per cubic centimetre |
| n_e | variable | Electron number density — electron number density per cubic metre |
| l | variable | Path length along line of sight — length metre |
| ∫ | operator | Integration along the line of sight — integral |
| kappa | variable | weak lensing convergence — convergence |
| theta | variable | angular position on the sky — angular position radian |
| chi | variable | comoving distance to the source — comoving distance metre |
| chi' | variable | comoving distance along the line of sight — comoving distance metre |
| q | variable | lensing efficiency function — lensing efficiency |
| a | variable | scale factor — scale factor |
| f_K | variable | comoving angular diameter distance — comoving angular diameter distance metre |
| delta | variable | matter density contrast — density contrast |
| H_0 | constant | current expansion rate of the universe — hubble constant = 2.2e-18 per second |
| Omega_m | constant | ratio of matter density to critical density today — matter density parameter = 0.315 |
| c | constant | speed of light in vacuum — speed of light = 299792458 metre per second |
| ∫ | operator | summation over infinitesimal elements — integral |