Editing Integral (section)

== Applications ==
Integrals are used extensively in many areas. For example, in [[probability theory]], integrals are used to determine the probability of some [[random variable]] falling within a certain range.<ref>{{Harvnb|Feller|1966|p=1}}.</ref> Moreover, the integral under an entire [[probability density function]] must equal 1, which provides a test of whether a [[Function (mathematics)|function]] with no negative values could be a density function or not.<ref>{{Harvnb|Feller|1966|p=3}}.</ref>

Integrals can be used for computing the [[area]] of a two-dimensional region that has a curved boundary, as well as [[Volume integral|computing the volume]] of a three-dimensional object that has a curved boundary. The area of a two-dimensional region can be calculated using the aforementioned definite integral.<ref>{{Harvnb|Apostol|1967|pp=88–89}}.</ref> The volume of a three-dimensional object such as a disc or washer can be computed by [[disc integration]] using the equation for the volume of a cylinder, <math>\pi r^2 h </math>, where <math>r</math> is the radius. In the case of a simple disc created by rotating a curve about the {{Math|''x''}}-axis, the radius is given by {{Math|''f''(''x'')}}, and its height is the differential {{Math|''dx''}}. Using an integral with bounds {{Math|''a''}} and {{Math|''b''}}, the volume of the disc is equal to:<ref>{{Harvnb|Apostol|1967|pp=111–114}}.</ref><math display="block">\pi \int_a^b f^2 (x) \, dx.</math>Integrals are also used in physics, in areas like [[kinematics]] to find quantities like [[Displacement (vector)|displacement]], [[time]], and [[velocity]]. For example, in [[rectilinear motion]], the displacement of an object over the time interval <math>[a,b]</math> is given by
: <math>x(b)-x(a) = \int_a^b v(t) \,dt,</math>

where <math>v(t)</math> is the velocity expressed as a function of time.<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=306}}.</ref> The work done by a force <math>F(x)</math> (given as a function of position) from an initial position <math>A</math> to a final position <math>B</math> is:<ref>{{Harvnb|Apostol|1967|p=116}}.</ref>

: <math>W_{A\rightarrow B} = \int_A^B F(x)\,dx.</math>

Integrals are also used in [[thermodynamics]], where [[thermodynamic integration]] is used to calculate the difference in free energy between two given states.