Definite Integrals

An integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

Integration is one of the two main operations of calculus; its inverse operation, differentiation, is the other.

Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral can be interpreted informally as the signed area of the region in the xy-plane that is bounded by the graph of the f, the x-axis, and the vertical lines x = a and x = b. The area above the x-axis adds to the total and the area below the x-axis subtracts from the total.

It is the fundamental theorem of calculus that connects differentiation with the definite integral: if f is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of f is known, the definite integral of f over that interval is given by the equation below.