integral
Sign in to savethumb|300px|A definite integral of a function can be represented as the signed area of the region bounded by its graph and the horizontal axis; in the above graph as an example, the integral of f(x) between a and b is the yellow (−) area subtracted from the blue (+) area|alt=Definite integral example
AI overview
An integral is a mathematical tool that calculates the area under a curve on a graph, accounting for whether regions are above or below the horizontal axis. It matters because it allows mathematicians and scientists to find total quantities like distance, area, and volume from information about how things change.
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Article
43 sectionsContents
- History
- Pre-calculus integration
- Leibniz and Newton
- Formalization
- Historical notation
- First use of the term
- Terminology and notation
- Interpretations
- Formal definitions
- Riemann integral
- Lebesgue integral
- Other integrals
- Properties
- Linearity
- Inequalities
- Conventions
- Fundamental theorem of calculus
- First theorem
- Second theorem
- Extensions
- Improper integrals
- Multiple integration
- Line integrals and surface integrals
- Contour integrals
- Integrals of differential forms
- Summations
- Functional integrals
- Applications
- Computation
- Analytical
- Symbolic
- Numerical
- Mechanical
- Geometrical
- Integration by differentiation
- Examples
- Using the fundamental theorem of calculus
- See also
- Notes
- References
- Bibliography
- External links
- Online books
thumb|300px|A definite integral of a function can be represented as the signed area of the region bounded by its graph and the horizontal axis; in the above graph as an example, the integral of f(x) between a and b is the yellow (−) area subtracted from the blue (+) area|alt=Definite integral example
In mathematics, an integral is the continuous analog of a sum, and is used to calculate areas, volumes, and their generalizations. The process of computing an integral, called integration, is one of the two fundamental operations of calculus, along with differentiation. Integration was initially used to solve problems in mathematics and physics, 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.