Integroimisen

Integroimisen, meaning the process of integration, is the mathematical operation of computing an integral. In Finnish mathematics the term covers both indefinite integration (finding an antiderivative) and definite integration (evaluating over an interval). The standard notation uses the integral sign ∫, with ∫ f(x) dx representing an antiderivative of f and ∫_a^b f(x) dx the accumulated quantity between a and b.

The Fundamental Theorem of Calculus links integroiminen and differentiation. Under suitable conditions, the derivative of the integral of a function equals the function, and the integral of a function's derivative recovers the original function up to a constant. This theorem explains how accumulation and rate of change are related.

Methods of integroiminen are analytic and numerical. Analytic techniques seek exact antiderivatives using substitutions, integration by parts, partial fractions, and special functions. Many integrals have no elementary antiderivative. When an exact form is unavailable, numerical integration estimates the value of definite integrals using methods such as the trapezoidal rule, Simpson's rule, or Gaussian quadrature.

Applications span area and volume computation, probability theory (expected value and moments), physics (work and energy), engineering, and economics. Integroimisen also underpins numerical methods in scientific computing.

Historically, the concept arose with Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century and was later formalized by Cauchy, Lebesgue, and others. Today integroiminen is a standard tool in calculus and mathematical analysis, taught in schools and used in research and computation.