# 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.