Integral: Difference between revisions
Jump to navigation
Jump to search
imported>OAbot m Open access bot: url-access=subscription updated in citation with #oabot. |
imported>SuperWikiaBros Redirected link and reviewed other links |
||
| Line 1: | Line 1: | ||
{{Short description|Operation in mathematical calculus}} | {{Short description|Operation in mathematical calculus}} | ||
{{hatnote group| | |||
{{About|the concept of definite integrals in calculus|the indefinite integral|antiderivative|the set of numbers|integer|other uses|Integral (disambiguation)}} | {{About|the concept of definite integrals in calculus|the indefinite integral|antiderivative|the set of numbers|integer|other uses|Integral (disambiguation)}} | ||
{{Redirect|Area under the curve | {{Redirect|Area under the curve}} | ||
[[File:Integral example.svg|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 <math>f(x)</math> is the yellow (−) area subtracted from the blue (+) area|alt=Definite integral example]] | }} | ||
[[File:Integral example.svg|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 <math>f(x)</math> between <math>a</math> and <math>b</math> is the yellow (−) area subtracted from the blue (+) area|alt=Definite integral example]] | |||
{{Calculus|Integral}} | {{Calculus|Integral}} | ||
In [[mathematics]], an '''integral''' is the continuous analog of a [[Summation|sum]], | In [[mathematics]], an '''integral''' is the continuous analog of a [[Summation|sum]], and is used to calculate [[area|areas]], [[volume|volumes]], and their generalizations. The process of computing an integral, called '''integration''', is one of the two fundamental operations of [[calculus]], along with [[Derivative|differentiation]].<ref group="lower-alpha">Integral calculus is a very well-established mathematical discipline for which there are many sources. See {{Harvnb|Apostol|1967}} and {{Harvnb|Anton|Bivens|Davis|2016}}, for example.</ref> Integration was initially used to solve problems in mathematics and [[physics]], such as finding the '''area under a curve''', or determining [[Displacement (geometry)|displacement]] from [[velocity]]. Usage of integration expanded to a wide variety of scientific fields thereafter. | ||
A '''definite integral''' computes the [[signed area]] of the region in the plane that is bounded by the [[Graph of a function|graph]] of a given [[Function (mathematics)|function]] between two points in the [[real line]]. Conventionally, areas above the horizontal [[Coordinate axis|axis]] of the plane are positive while areas below are negative. Integrals also refer to the concept of an ''[[antiderivative]]'', a function whose [[derivative]] is the given function; in this case, they are also called ''indefinite integrals''. The [[fundamental theorem of calculus]] 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 [[inverse function|inverse]] operations. | A '''definite integral''' computes the [[signed area]] of the region in the plane that is bounded by the [[Graph of a function|graph]] of a given [[Function (mathematics)|function]] between two points in the [[real line]]. Conventionally, areas above the horizontal [[Coordinate axis|axis]] of the plane are positive while areas below are negative. Integrals also refer to the concept of an ''[[antiderivative]]'', a function whose [[derivative]] is the given function; in this case, they are also called ''indefinite integrals''. The [[fundamental theorem of calculus]] 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 [[inverse function|inverse]] operations. | ||
| Line 17: | Line 19: | ||
=== Pre-calculus integration === | === Pre-calculus integration === | ||
The first documented systematic technique capable of determining integrals is the [[method of exhaustion]] of the [[Ancient | The first documented systematic technique capable of determining integrals is the [[method of exhaustion]] of the [[Ancient Greek mathematics|ancient Greek]] astronomer [[Eudoxus of Cnidus|Eudoxus]] and philosopher [[Democritus]] (''ca.'' 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known.<ref>{{Harvnb|Burton|2011|p=117}}.</ref> This method was further developed and employed by [[Archimedes]] in the 3rd century BC and used to calculate the [[area of a circle]], the [[surface area]] and [[volume]] of a [[sphere]], area of an [[ellipse]], the area under a [[parabola]], the volume of a segment of a [[paraboloid]] of revolution, the volume of a segment of a [[hyperboloid]] of revolution, and the area of a [[spiral]].<ref>{{Harvnb|Heath|2002}}.</ref> | ||
A similar method was independently developed in [[China]] around the 3rd century AD by [[Liu Hui]], who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians [[Zu Chongzhi]] and [[Zu Geng (mathematician)|Zu Geng]] to find the volume of a sphere.<ref>{{harvnb|Katz|2009|pp=201–204}}.</ref> | A similar method was independently developed in [[Chinese mathematics|China]] around the 3rd century AD by [[Liu Hui]], who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians [[Zu Chongzhi]] and [[Zu Geng (mathematician)|Zu Geng]] to find the volume of a sphere.<ref>{{harvnb|Katz|2009|pp=201–204}}.</ref> | ||
In the Middle East, Hasan Ibn al-Haytham, Latinized as [[Alhazen]] ({{c.|965|lk=no|1040}} AD) derived a formula for the sum of [[fourth power]]s.<ref>{{harvnb|Katz|2009|pp=284–285}}.</ref> Alhazen determined the equations to calculate the area enclosed by the curve represented by <math>y=x^k</math> (which translates to the integral <math>\int x^k \, dx</math> in contemporary notation), for any given non-negative integer value of <math>k</math>.<ref>{{Cite journal | | In the Middle East, Hasan Ibn al-Haytham, Latinized as [[Alhazen]] ({{c.|965|lk=no|1040}} AD) derived a formula for the sum of [[fourth power]]s.<ref>{{harvnb|Katz|2009|pp=284–285}}.</ref> Alhazen determined the equations to calculate the area enclosed by the curve represented by <math>y=x^k</math> (which translates to the integral <math>\int x^k \, dx</math> in contemporary notation), for any given non-negative integer value of <math>k</math>.<ref>{{Cite journal |last1=Dennis |first1=David |last2=Kreinovich |first2=Vladik |last3=Rump |first3=Siegfried M. |date=1998-05-01 |title=Intervals and the Origins of Calculus |journal=Reliable Computing |language=en |volume=4 |issue=2 |pages=191–197 |doi=10.1023/A:1009989211143 |issn=1573-1340}}</ref> He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a [[paraboloid]].<ref>{{harvnb|Katz|2009|pp=305–306}}.</ref> | ||
The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of [[Bonaventura Cavalieri|Cavalieri]] with his [[method of indivisibles]], and work by [[Pierre de Fermat|Fermat]], began to lay the foundations of modern calculus,<ref>{{harvnb|Katz|2009|pp=516–517}}.</ref> with Cavalieri computing the integrals of {{math|''x''<sup>''n''</sup>}} up to degree {{math|''n'' {{=}} 9}} in [[Cavalieri's quadrature formula]].<ref>{{Harvnb|Struik|1986|pp=215–216}}.</ref> The case ''n'' = −1 required the invention of a [[function (mathematics)|function]], the [[hyperbolic logarithm]], achieved by [[quadrature (mathematics)|quadrature]] of the [[hyperbola]] in 1647. | The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of [[Bonaventura Cavalieri|Cavalieri]] with his [[method of indivisibles]], and work by [[Pierre de Fermat|Fermat]], began to lay the foundations of modern calculus,<ref>{{harvnb|Katz|2009|pp=516–517}}.</ref> with Cavalieri computing the integrals of {{math|''x''<sup>''n''</sup>}} up to degree {{math|''n'' {{=}} 9}} in [[Cavalieri's quadrature formula]].<ref>{{Harvnb|Struik|1986|pp=215–216}}.</ref> The case ''n'' = −1 required the invention of a [[function (mathematics)|function]], the [[hyperbolic logarithm]], achieved by [[quadrature (mathematics)|quadrature]] of the [[hyperbola]] in 1647. | ||
| Line 44: | Line 46: | ||
In general, the integral of a [[real-valued function]] {{Math|1=''f''(''x'')}} with respect to a real variable {{Math|1=''x''}} on an interval {{Math|1=[''a'', ''b'']}} is written as | In general, the integral of a [[real-valued function]] {{Math|1=''f''(''x'')}} with respect to a real variable {{Math|1=''x''}} on an interval {{Math|1=[''a'', ''b'']}} is written as | ||
:<math>\int_{a}^{b} f(x) \,dx.</math> | :<math>\int_{a}^{b} f(x) \,dx.</math> | ||
The integral | The [[integral symbol]] {{Math|∫}} represents integration. The symbol {{Math|''dx''}}, called the [[Differential (infinitesimal)|differential]] of the variable {{Math|1=''x''}}, indicates that the variable of integration is {{Math|1=''x''}}. The function {{Math|1=''f''(''x'')}} is called the ''integrand'', the points {{Math|1=''a''}} and {{Math|1=''b''}} are called the limits (or bounds) of integration, and the integral is said to be over the interval {{Math|1=[''a'', ''b'']}}, called the interval of integration.<ref name=":1">{{Harvnb|Apostol|1967|p=74}}.</ref> | ||
A function is said to be {{em|integrable}}{{anchor|Integrable|Integrable function}} if its integral over its domain is finite. If limits are specified, the integral is called a definite integral. | A function is said to be {{em|integrable}}{{anchor|Integrable|Integrable function}} if its integral over its domain is finite. If limits are specified, the integral is called a definite integral. | ||
| Line 98: | Line 100: | ||
| alt1 = Riemann integral approximation example | | alt1 = Riemann integral approximation example | ||
| caption1 = Integral example with irregular partitions (largest marked in red) | | caption1 = Integral example with irregular partitions (largest marked in red) | ||
| image2 = Riemann sum convergence. | | image2 = Riemann sum convergence.svg | ||
| alt2 = Riemann sum convergence | | alt2 = Riemann sum convergence | ||
| caption2 = Riemann sums converging | | caption2 = Riemann sums converging | ||
| Line 136: | Line 138: | ||
: <math>\int f = \int_0^\infty f^*(t)\,dt</math> | : <math>\int f = \int_0^\infty f^*(t)\,dt</math> | ||
where the integral on the right is an ordinary improper Riemann integral ({{math|''f''{{i sup|∗}}}} is a strictly decreasing positive function, and therefore has a | where the integral on the right is an ordinary improper Riemann integral ({{math|''f''{{i sup|∗}}}} is a strictly decreasing positive function, and therefore has a well-defined improper Riemann integral).<ref>{{Harvnb|Lieb|Loss|2001|p=14}}.</ref> For a suitable class of functions (the [[measurable function]]s) this defines the Lebesgue integral. | ||
A general measurable function {{mvar|f}} is Lebesgue-integrable if the sum of the absolute values of the areas of the regions between the graph of {{mvar|f}} and the {{mvar|x}}-axis is finite:<ref>{{Harvnb|Folland|1999|p=53}}.</ref> | A general measurable function {{mvar|f}} is Lebesgue-integrable if the sum of the absolute values of the areas of the regions between the graph of {{mvar|f}} and the {{mvar|x}}-axis is finite:<ref>{{Harvnb|Folland|1999|p=53}}.</ref> | ||
| Line 172: | Line 174: | ||
* The [[Young integral]], which is a kind of Riemann–Stieltjes integral with respect to certain functions of [[Bounded variation|unbounded variation]]. | * The [[Young integral]], which is a kind of Riemann–Stieltjes integral with respect to certain functions of [[Bounded variation|unbounded variation]]. | ||
* The [[rough path]] integral, which is defined for functions equipped with some additional "rough path" structure and generalizes stochastic integration against both [[semimartingale]]s and processes such as the [[fractional Brownian motion]]. | * The [[rough path]] integral, which is defined for functions equipped with some additional "rough path" structure and generalizes stochastic integration against both [[semimartingale]]s and processes such as the [[fractional Brownian motion]]. | ||
* The [[Choquet integral]], a subadditive or superadditive integral created by | * The [[Choquet integral]], a subadditive or superadditive integral created by [[Gustave Choquet]] in 1953. | ||
* The [[Bochner integral]], a generalization of the Lebesgue integral to functions that take values in a [[Banach space]]. | * The [[Bochner integral]], a generalization of the Lebesgue integral to functions that take values in a [[Banach space]]. | ||
| Line 326: | Line 328: | ||
: <math>\int_S {\mathbf v}\cdot \,d{\mathbf S}.</math> | : <math>\int_S {\mathbf v}\cdot \,d{\mathbf S}.</math> | ||
The fluid flux in this example may be from a physical fluid such as water or air, or from electrical or magnetic flux. Thus surface integrals have applications in physics, particularly with the [[classical theory | The fluid flux in this example may be from a physical fluid such as water or air, or from electrical or magnetic flux. Thus surface integrals have applications in physics, particularly with the [[Electromagnetism|classical theory of electromagnetism]]. | ||
=== Contour integrals === | === Contour integrals === | ||
| Line 403: | Line 405: | ||
===Numerical=== | ===Numerical=== | ||
{{Main|Numerical integration}} | {{Main|Numerical integration}} | ||
[[File:Numerical_quadrature_4up.png|right|thumb|Numerical quadrature methods: rectangle method, trapezoidal rule, Romberg's method, Gaussian quadrature]] | [[File:Numerical_quadrature_4up.png|right|thumb|Numerical quadrature methods: rectangle method, trapezoidal rule, Romberg's method (at varying piece count), Gaussian quadrature]] | ||
Definite integrals may be approximated using several methods of [[numerical integration]]. The [[rectangle method]] relies on dividing the region under the function into a series of rectangles corresponding to function values and multiplies by the step width to find the sum. A better approach, the [[trapezoidal rule]], replaces the rectangles used in a Riemann sum with trapezoids. The trapezoidal rule weights the first and last values by one half, then multiplies by the step width to obtain a better approximation.<ref>{{Harvnb|Dahlquist|Björck|2008|pp=519–520}}.</ref> The idea behind the trapezoidal rule, that more accurate approximations to the function yield better approximations to the integral, can be carried further: [[Simpson's rule]] approximates the integrand by a piecewise quadratic function.<ref>{{Harvnb|Dahlquist|Björck|2008|pp=522–524}}.</ref> | Definite integrals may be approximated using several methods of [[numerical integration]]. The [[rectangle method]] relies on dividing the region under the function into a series of rectangles corresponding to function values and multiplies by the step width to find the sum. A better approach, the [[trapezoidal rule]], replaces the rectangles used in a Riemann sum with trapezoids. The trapezoidal rule weights the first and last values by one half, then multiplies by the step width to obtain a better approximation.<ref>{{Harvnb|Dahlquist|Björck|2008|pp=519–520}}.</ref> The idea behind the trapezoidal rule, that more accurate approximations to the function yield better approximations to the integral, can be carried further: [[Simpson's rule]] approximates the integrand by a piecewise quadratic function.<ref>{{Harvnb|Dahlquist|Björck|2008|pp=522–524}}.</ref> | ||
| Line 444: | Line 446: | ||
{{refbegin|35em}} | {{refbegin|35em}} | ||
* {{Citation |last1=Anton |first1=Howard |last2=Bivens |first2=Irl C. |last3=Davis |first3=Stephen |title=Calculus: Early Transcendentals |volume= |pages= |year=2016 |edition=11th |publisher=John Wiley & Sons |isbn=978-1-118-88382-2}} | * {{Citation |last1=Anton |first1=Howard |last2=Bivens |first2=Irl C. |last3=Davis |first3=Stephen |title=Calculus: Early Transcendentals |volume= |pages= |year=2016 |edition=11th |publisher=John Wiley & Sons |isbn=978-1-118-88382-2}} | ||
* {{Citation |last=Apostol |first=Tom M. |title=Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra | | * {{Citation |last=Apostol |first=Tom M. |title=Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra |title-link=Calculus (Apostol books) |volume= |pages= |year=1967 |edition=2nd |publisher=Wiley |isbn=978-0-471-00005-1 |author-link=Tom M. Apostol}} | ||
* {{Citation |last=Bourbaki |first=Nicolas |title=Integration I |volume= |pages= |year=2004 |publisher=Springer-Verlag |isbn=3-540-41129-1 |author-link=Nicolas Bourbaki}}. In particular chapters III and IV. | * {{Citation |last=Bourbaki |first=Nicolas |title=Integration I |volume= |pages= |year=2004 |publisher=Springer-Verlag |isbn=3-540-41129-1 |author-link=Nicolas Bourbaki}}. In particular chapters III and IV. | ||
* {{Citation |last=Burton |first=David M. |title=The History of Mathematics: An Introduction |volume= |pages= |year=2011 |edition=7th |publisher=McGraw-Hill |isbn=978-0-07-338315-6}} | * {{Citation |last=Burton |first=David M. |title=The History of Mathematics: An Introduction |volume= |pages= |year=2011 |edition=7th |publisher=McGraw-Hill |isbn=978-0-07-338315-6}} | ||
* {{Citation |last=Cajori |first=Florian |title=A History Of Mathematical Notations Volume II |url=https://archive.org/details/historyofmathema00cajo_0/page/247 |volume= |pages= |year=1929 |publisher=Open Court Publishing |isbn=978-0-486-67766-8 |author-link=Florian Cajori}} | * {{Citation |last=Cajori |first=Florian |title=A History Of Mathematical Notations Volume II |url=https://archive.org/details/historyofmathema00cajo_0/page/247 |volume= |pages= |year=1929 |publisher=Open Court Publishing |isbn=978-0-486-67766-8 |author-link=Florian Cajori}} | ||
* {{Citation |last1=Dahlquist |first1=Germund |title=Numerical Methods in Scientific Computing, Volume I |volume= |pages= |year=2008 |archive-url=https://web.archive.org/web/20070615185623/http://www.mai.liu.se/~akbjo/NMbook.html |chapter=Chapter 5: Numerical Integration |chapter-url=http://www.mai.liu.se/~akbjo/NMbook.html |location=Philadelphia |publisher=[[Society for Industrial and Applied Mathematics|SIAM]] |archive-date=2007-06-15 |last2=Björck |first2=Åke |author1-link=Germund Dahlquist | * {{Citation |last1=Dahlquist |first1=Germund |title=Numerical Methods in Scientific Computing, Volume I |volume= |pages= |year=2008 |archive-url=https://web.archive.org/web/20070615185623/http://www.mai.liu.se/~akbjo/NMbook.html |chapter=Chapter 5: Numerical Integration |chapter-url=http://www.mai.liu.se/~akbjo/NMbook.html |location=Philadelphia |publisher=[[Society for Industrial and Applied Mathematics|SIAM]] |archive-date=2007-06-15 |last2=Björck |first2=Åke |author1-link=Germund Dahlquist }} | ||
* {{Citation |last=Feller |first=William |author-link=William Feller |title=An introduction to probability theory and its applications |pages= |year=1966 |publisher=John Wiley & Sons |url=https://archive.org/details/introductiontopr02fell_0 |volume= |url-access=registration}} | * {{Citation |last=Feller |first=William |author-link=William Feller |title=An introduction to probability theory and its applications |pages= |year=1966 |publisher=John Wiley & Sons |url=https://archive.org/details/introductiontopr02fell_0 |volume= |url-access=registration}} | ||
* {{Citation |last=Folland |first=Gerald B. |author-link=Gerald Folland |title=Real Analysis: Modern Techniques and Their Applications |volume= |pages= |year=1999 |edition=2nd |publisher=John Wiley & Sons |isbn=0-471-31716-0}} | * {{Citation |last=Folland |first=Gerald B. |author-link=Gerald Folland |title=Real Analysis: Modern Techniques and Their Applications |volume= |pages= |year=1999 |edition=2nd |publisher=John Wiley & Sons |isbn=0-471-31716-0}} | ||
| Line 456: | Line 458: | ||
* {{Citation |last=Hildebrandt |first=T. H. |title=Integration in abstract spaces |url=http://projecteuclid.org/euclid.bams/1183517761 |journal=[[Bulletin of the American Mathematical Society]] |volume=59 |issue=2 |pages=111–139 |year=1953 |doi=10.1090/S0002-9904-1953-09694-X |issn=0273-0979 |author-link=Theophil Henry Hildebrandt |doi-access=free}} | * {{Citation |last=Hildebrandt |first=T. H. |title=Integration in abstract spaces |url=http://projecteuclid.org/euclid.bams/1183517761 |journal=[[Bulletin of the American Mathematical Society]] |volume=59 |issue=2 |pages=111–139 |year=1953 |doi=10.1090/S0002-9904-1953-09694-X |issn=0273-0979 |author-link=Theophil Henry Hildebrandt |doi-access=free}} | ||
* {{Citation |last1=Kahaner |first1=David |title=Numerical Methods and Software |url=https://archive.org/details/numericalmethods0000kaha |volume= |pages= |year=1989 |chapter=Chapter 5: Numerical Quadrature |publisher=Prentice Hall |isbn=978-0-13-627258-8 |last2=Moler |first2=Cleve |last3=Nash |first3=Stephen |author2-link=Cleve Moler |url-access=registration}} | * {{Citation |last1=Kahaner |first1=David |title=Numerical Methods and Software |url=https://archive.org/details/numericalmethods0000kaha |volume= |pages= |year=1989 |chapter=Chapter 5: Numerical Quadrature |publisher=Prentice Hall |isbn=978-0-13-627258-8 |last2=Moler |first2=Cleve |last3=Nash |first3=Stephen |author2-link=Cleve Moler |url-access=registration}} | ||
*{{Citation |last=Kallio |first=Bruce Victor |title=A History of the Definite Integral |url=https://open.library.ubc.ca/soa/cIRcle/collections/ubctheses/831/items/1.0080597 |volume= |pages= |year=1966 |type=M.A. thesis |archive-url=https://web.archive.org/web/20140305054035/https://circle.ubc.ca/bitstream/id/132341/UBC_1966_A8%20K3.pdf |publisher=University of British Columbia |access-date=2014-02-28 |archive-date=2014-03-05 | *{{Citation |last=Kallio |first=Bruce Victor |title=A History of the Definite Integral |url=https://open.library.ubc.ca/soa/cIRcle/collections/ubctheses/831/items/1.0080597 |volume= |pages= |year=1966 |type=M.A. thesis |archive-url=https://web.archive.org/web/20140305054035/https://circle.ubc.ca/bitstream/id/132341/UBC_1966_A8%20K3.pdf |publisher=University of British Columbia |doi=10.14288/1.0080597 |access-date=2014-02-28 |archive-date=2014-03-05 }} | ||
*{{Citation |last=Katz |first=Victor J. |author-link=Victor J. Katz |title=A History of Mathematics: An Introduction |volume= |pages= |year=2009 |publisher=[[Addison-Wesley]] |isbn=978-0-321-38700-4}} | *{{Citation |last=Katz |first=Victor J. |author-link=Victor J. Katz |title=A History of Mathematics: An Introduction |volume= |pages= |year=2009 |publisher=[[Addison-Wesley]] |isbn=978-0-321-38700-4}} | ||
*{{Citation |last1=Kempf |first1=Achim |last2=Jackson |first2=David M. |last3=Morales |first3=Alejandro H. |title=How to (path-)integrate by differentiating |journal=Journal of Physics: Conference Series |volume=626 | | *{{Citation |last1=Kempf |first1=Achim |last2=Jackson |first2=David M. |last3=Morales |first3=Alejandro H. |title=How to (path-)integrate by differentiating |journal=Journal of Physics: Conference Series |volume=626 |article-number=012015 |year=2015 |issue=1 |publisher=[[IOP Publishing]] |doi=10.1088/1742-6596/626/1/012015 |arxiv=1507.04348 |bibcode=2015JPhCS.626a2015K |s2cid=119642596}} | ||
* {{Citation |last=Krantz |first=Steven G. |title=Real Analysis and Foundations |year=1991 |publisher=CRC Press |author-link=Steven G. Krantz |url=https://books.google.com/books?id=OI-0vu1rb7MC&pg=PA173 |isbn=0-8493-7156-2}} | * {{Citation |last=Krantz |first=Steven G. |title=Real Analysis and Foundations |year=1991 |publisher=CRC Press |author-link=Steven G. Krantz |url=https://books.google.com/books?id=OI-0vu1rb7MC&pg=PA173 |isbn=0-8493-7156-2}} | ||
* {{Citation |last=Leibniz |first=Gottfried Wilhelm |title=Der Briefwechsel von Gottfried Wilhelm Leibniz mit Mathematikern. Erster Band |url=http://name.umdl.umich.edu/AAX2762.0001.001 |volume= |pages= |year=1899 |editor-last=Gerhardt |editor-first=Karl Immanuel |place=Berlin |publisher=Mayer & Müller |author-link=Gottfried Wilhelm Leibniz}} | * {{Citation |last=Leibniz |first=Gottfried Wilhelm |title=Der Briefwechsel von Gottfried Wilhelm Leibniz mit Mathematikern. Erster Band |url=http://name.umdl.umich.edu/AAX2762.0001.001 |volume= |pages= |year=1899 |editor-last=Gerhardt |editor-first=Karl Immanuel |place=Berlin |publisher=Mayer & Müller |author-link=Gottfried Wilhelm Leibniz}} | ||
* {{citation |last1=Lieb |first1=Elliott |title=Analysis |volume=14 |pages= |year=2001 |series=[[Graduate Studies in Mathematics]] |edition=2nd |publisher=[[American Mathematical Society]] |isbn=978- | * {{citation |last1=Lieb |first1=Elliott |title=Analysis |volume=14 |pages= |year=2001 |series=[[Graduate Studies in Mathematics]] |edition=2nd |publisher=[[American Mathematical Society]] |isbn=978-0-8218-2783-3 |last2=Loss |first2=Michael |author-link1=Elliott H. Lieb |author2-link=Michael Loss}} | ||
* {{citation |last1=Montesinos |first1=Vicente |last2=Zizler |first2=Peter |last3=Zizler |first3=Václav |title=An Introduction to Modern Analysis |edition=illustrated |publisher=Springer |year=2015 |isbn=978-3-319-12481-0 |url=https://books.google.com/books?id=mlX1CAAAQBAJ&pg=PA355}} | * {{citation |last1=Montesinos |first1=Vicente |last2=Zizler |first2=Peter |last3=Zizler |first3=Václav |title=An Introduction to Modern Analysis |edition=illustrated |publisher=Springer |year=2015 |isbn=978-3-319-12481-0 |url=https://books.google.com/books?id=mlX1CAAAQBAJ&pg=PA355}} | ||
* Paul J. Nahin (2015), ''Inside Interesting Integrals'', Springer, ISBN 978-1-4939-1276-6. | * Paul J. Nahin (2015), ''Inside Interesting Integrals'', Springer, ISBN 978-1-4939-1276-6. | ||
*{{Citation |last1=Rich |first1=Albert |title=Rule-based integration: An extensive system of symbolic integration rules |date=16 December 2018 |journal=Journal of Open Source Software |volume=3 |issue=32 | | *{{Citation |last1=Rich |first1=Albert |title=Rule-based integration: An extensive system of symbolic integration rules |date=16 December 2018 |journal=Journal of Open Source Software |volume=3 |issue=32 |page=1073 |doi=10.21105/joss.01073 |last2=Scheibe |first2=Patrick |last3=Abbasi |first3=Nasser |bibcode=2018JOSS....3.1073R |s2cid=56487062 |doi-access=free}} | ||
* {{Citation |last=Rudin |first=Walter |title=Real and Complex Analysis |volume= |pages= |year=1987 |chapter=Chapter 1: Abstract Integration |edition=International |publisher=McGraw-Hill |isbn=978-0-07-100276-9 |author-link=Walter Rudin}} | * {{Citation |last=Rudin |first=Walter |title=Real and Complex Analysis |volume= |pages= |year=1987 |chapter=Chapter 1: Abstract Integration |edition=International |publisher=McGraw-Hill |isbn=978-0-07-100276-9 |author-link=Walter Rudin}} | ||
* {{Citation |last=Saks |first=Stanisław |title=Theory of the integral |url=http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez= |volume= |pages= |year=1964 |edition=English translation by L. C. Young. With two additional notes by Stefan Banach. Second revised |place=New York |publisher=Dover |author-link=Stanisław Saks}} | * {{Citation |last=Saks |first=Stanisław |title=Theory of the integral |url=http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez= |volume= |pages= |year=1964 |edition=English translation by L. C. Young. With two additional notes by Stefan Banach. Second revised |place=New York |publisher=Dover |author-link=Stanisław Saks}} | ||
| Line 489: | Line 491: | ||
* Hussain, Faraz, [http://www.understandingcalculus.com Understanding Calculus], an online textbook | * Hussain, Faraz, [http://www.understandingcalculus.com Understanding Calculus], an online textbook | ||
* Johnson, William Woolsey (1909) [http://babel.hathitrust.org/cgi/pt?id=miun.aam9447.0001.001;view=1up;seq=9 Elementary Treatise on Integral Calculus], link from [[HathiTrust]]. | * Johnson, William Woolsey (1909) [http://babel.hathitrust.org/cgi/pt?id=miun.aam9447.0001.001;view=1up;seq=9 Elementary Treatise on Integral Calculus], link from [[HathiTrust]]. | ||
* Kowalk, W. P., [http://einstein.informatik.uni-oldenburg.de/20910.html ''Integration Theory''], University of Oldenburg. A new concept to an old problem. Online textbook | * Kowalk, W. P., [http://einstein.informatik.uni-oldenburg.de/20910.html ''Integration Theory''] {{Webarchive|url=https://web.archive.org/web/20120227032738/http://einstein.informatik.uni-oldenburg.de/20910.html |date=2012-02-27 }}, University of Oldenburg. A new concept to an old problem. Online textbook | ||
* Sloughter, Dan, [http://math.furman.edu/~dcs/book Difference Equations to Differential Equations], an introduction to calculus | * Sloughter, Dan, [http://math.furman.edu/~dcs/book Difference Equations to Differential Equations] {{Webarchive|url=https://web.archive.org/web/20110715014621/http://math.furman.edu/~dcs/book/ |date=2011-07-15 }}, an introduction to calculus | ||
* [http://numericalmethods.eng.usf.edu/topics/integration.html Numerical Methods of Integration] at ''Holistic Numerical Methods Institute'' | * [http://numericalmethods.eng.usf.edu/topics/integration.html Numerical Methods of Integration] at ''Holistic Numerical Methods Institute'' | ||
* P. S. Wang, [https://web.archive.org/web/20060917023831/http://www.lcs.mit.edu/publications/specpub.php?id=660 Evaluation of Definite Integrals by Symbolic Manipulation] (1972) — a cookbook of definite integral techniques | * P. S. Wang, [https://web.archive.org/web/20060917023831/http://www.lcs.mit.edu/publications/specpub.php?id=660 Evaluation of Definite Integrals by Symbolic Manipulation] (1972) — a cookbook of definite integral techniques | ||