Numerical integration

Numerical integration approximates definite integrals using finite computations, typically by combining function values over subintervals. It is used when exact antiderivatives are unavailable or computation is more practical.

Riemann sum: An approximation to an integral formed by summing function values multiplied by subinterval widths. This basic construction turns sampled heights into an area estimate.
Definite integral: The signed accumulation of a function over a specified interval or region. Numerical integration seeks an approximation to this quantity.
Ordinary differential equation: An equation relating an unknown function of one variable to its derivatives. Time-stepping solvers often approximate integrals of rates over each step.
Symbolic integration: The exact or symbolic determination of an integral as a formula or mathematical expression. It can produce closed forms where numerical integration returns approximations.
Quadrature error: The difference between an integral and its approximation by a quadrature rule. Reliable error estimates determine whether additional evaluations are needed.
Trapezoidal rule: A quadrature rule that approximates a function by straight segments between sample points. Its simple geometric estimate is exact for linear functions.
Numerical error: The difference between a computed numerical result and its exact or intended value. Integration methods are judged by how their errors behave.
Finite element method: A numerical method that approximates solutions to boundary-value problems using piecewise functions. Its element matrices and loads commonly require numerical evaluation of integrals.
Monte Carlo method: A broad class of computational methods that use random sampling to estimate quantities. Monte Carlo integration is one member of this larger family.
Curse of dimensionality: The rapid growth in computational resources needed as the dimension of a problem increases. Standard grids become prohibitively expensive for high-dimensional integration.

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