integrals

In calculus, an integral assigns a number to a function, representing accumulation. There are indefinite integrals, which yield antiderivatives, and definite integrals, which compute accumulated quantities over an interval. The standard notation is ∫ f(x) dx for an antiderivative and ∫_a^b f(x) dx for a definite integral. The Fundamental Theorem of Calculus links differentiation and integration by showing that the derivative of an antiderivative is the original function and that the definite integral equals the net change of this antiderivative over [a, b].

There are several ways to define and handle integrals. The Riemann integral partitions the domain and sums rectangle areas on subintervals. The Lebesgue integral, more general, measures the size of the sets where the function takes on certain values and can handle more irregular functions. Improper integrals extend the notion to infinite intervals or unbounded integrands.

Techniques for finding antiderivatives include substitution, integration by parts, partial fractions, and various trigonometric methods. Some functions have elementary antiderivatives, while others do not. When a closed form is unavailable, numerical methods provide approximate values.

Numerical methods for definite integrals include the trapezoidal rule, Simpson’s rule, and adaptive quadrature. These are essential in science and engineering for evaluating integrals that lack simple antiderivatives.

Applications of integrals span area calculations, probability (expected values and distributions), physics (work and energy), and economics (consumer surplus). Core properties include linearity and additivity: the integral of a sum equals the sum of the integrals, and constants factor out.

Historically, the concept emerged in the 17th century through the work of Newton and Leibniz and was later formalized by Cauchy and Lebesgue.