Integraaleja

In mathematics, integrals (integraaleja in Finnish) are a central concept in calculus that quantify accumulation and the area under a curve. An integral can be interpreted as the limit of sums of infinitesimal contributions, providing a way to measure quantities that accumulate continuously, such as distance, mass, or probability.

There are two main kinds: indefinite integrals and definite integrals. An indefinite integral, written as ∫ f(x) dx, represents the family of antiderivatives F(x) with F'(x) = f(x). A definite integral, written as ∫_a^b f(x) dx, yields a single number that represents the accumulated amount from x = a to x = b, often corresponding to the area between f and the x-axis (when f is nonnegative).

Integrals are evaluated using various techniques, including substitution, integration by parts, partial fractions, and special functions. For functions that resist elementary methods, numerical approaches such as the trapezoidal or Simpson's rules approximate the value of an integral. Improper integrals allow integration over infinite intervals or of unbounded functions, under convergence conditions.

Historically, the calculus of integrals developed in the 17th century by Isaac Newton and Gottfried Wilhelm Leibniz, building on earlier geometric ideas. Today integrals are essential across mathematics, physics, engineering, and statistics, with applications ranging from computing areas and probabilities to solving differential equations and modeling physical work.