Integral

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an integral is the continuous analog of a <a href="/facts/Summation/V2WUxoKn">sum</a>, which is used to calculate <a href="/facts/Area/KsNG1G9t">areas</a>, <a href="/facts/Volume/aeAXCBUd">volumes</a>, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of <a href="/facts/Calculus/MEmUTYoz">calculus</a>, the other being <a href="/facts/Derivative/7Ito70Dh">differentiation</a>. Integration was initially used to solve problems in mathematics and <a href="/facts/Physics/a7aH2y08">physics</a>, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter.
A definite integral computes the <a href="/facts/Signed_area/V7lyV997">signed area</a> of the region in the plane that is bounded by the <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> of a given <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> between two points in the <a href="/facts/Real_line/6eq42xX5">real line</a>. Conventionally, areas above the horizontal <a href="/facts/Coordinate_axis/Mtd0q4SP">axis</a> of the plane are positive while areas below are negative. Integrals also refer to the concept of an <a href="/facts/Antiderivative/5qDsudIv">antiderivative</a>, a function whose <a href="/facts/Derivative/7Ito70Dh">derivative</a> is the given function; in this case, they are also called indefinite integrals. The <a href="/facts/Fundamental_theorem_of_calculus/pHeSGtUx">fundamental theorem of calculus</a> relates definite integration to differentiation and provides a method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are <a href="/facts/Inverse_function/Tg5nnXlu">inverse</a> operations.
Although methods of calculating areas and volumes dated from <a href="/facts/Ancient_Greek_mathematics/ealke3z8">ancient Greek mathematics</a>, the principles of integration were formulated independently by <a href="/facts/Isaac_Newton/4L7gTncN">Isaac Newton</a> and <a href="/facts/Gottfried_Wilhelm_Leibniz/cTc2er83">Gottfried Wilhelm Leibniz</a> in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of <a href="/facts/Infinitesimal/JKcTMIgD">infinitesimal</a> width. <a href="/facts/Bernhard_Riemann/oV2fL0k8">Bernhard Riemann</a> later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a <a href="/facts/Curvilinear_coordinates/RQD7Cajd">curvilinear</a> region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century, <a href="/facts/Henri_Lebesgue/ysngVGXc">Henri Lebesgue</a> generalized Riemann's formulation by introducing what is now referred to as the <a href="/facts/Lebesgue_integration/f4AgvC6x">Lebesgue integral</a>; it is more general than Riemann's in the sense that a wider class of functions are Lebesgue-integrable.
Integrals may be generalized depending on the type of the function as well as the <a href="/facts/Domain_(mathematical_analysis)/xT44oI3E">domain</a> over which the integration is performed. For example, a <a href="/facts/Line_integral/10s4B3Nn">line integral</a> is defined for functions of two or more variables, and the <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> of integration is replaced by a curve connecting two points in space. In a <a href="/facts/Surface_integral/oPk5IRI1">surface integral</a>, the curve is replaced by a piece of a <a href="/facts/Surface_(mathematics)/whF08jvy">surface</a> in <a href="/facts/Three-dimensional_space/KAqKWUbS">three-dimensional space</a>.

Integral open-in-new

Integral