View source for Cavalieri's quadrature formula

{{distinguish|Cavalieri's principle}}
[[File:X cubed plot.svg|thumb|Cavalieri's quadrature formula computes the area under the [[cubic curve]], together with other higher powers.]]
In [[calculus]], '''Cavalieri's quadrature formula''', named for 17th-century Italian mathematician [[Bonaventura Cavalieri]], is the [[integral]]

:<math>\int_0^a x^n\,dx = \tfrac{1}{n+1}\, a^{n+1} \qquad n \geq 0,</math>

and generalizations thereof. This is the [[definite integral]] form; the [[indefinite integral]] form is:

:<math>\int x^n\,dx = \tfrac{1}{n+1}\, x^{n+1} + C \qquad n \neq -1.</math>

There are additional [[#Forms|forms]], listed below. Together with the [[linearity]] of the integral, this formula allows one to compute the integrals of all polynomials.

The term "[[Numerical integration|quadrature]]" is a traditional term for [[area]]; the integral is geometrically interpreted as the area under the curve ''y''&nbsp;=&nbsp;''x''<sup>''n''</sup>. Traditionally important cases are ''y''&nbsp;=&nbsp;''x''<sup>2</sup>, the quadrature of the [[parabola]], known in antiquity, and ''y''&nbsp;=&nbsp;1/''x'', the quadrature of the hyperbola, whose value is a [[logarithm]].

== Forms ==
=== Negative ''n'' ===
For negative values of ''n'' (negative powers of ''x''), there is a [[Mathematical singularity|singularity]] at ''x''&nbsp;=&nbsp;0, and thus the definite integral is based at 1, rather than 0, yielding:

:<math>\int_1^a x^n\,dx = \frac{1}{n+1} (a^{n+1} - 1) \qquad n \neq -1.</math>

Further, for negative fractional (non-integer) values of ''n,'' the power ''x''<sup>''n''</sup> is not [[well-defined]], hence the indefinite integral is only defined for positive ''x.'' However, for ''n'' a negative integer the power ''x''<sup>''n''</sup> is defined for all non-zero ''x,'' and the indefinite integrals and definite integrals are defined, and can be computed via a symmetry argument, replacing ''x'' by &minus;''x,'' and basing the negative definite integral at&nbsp;&minus;1.

Over the complex numbers the definite integral (for negative values of ''n'' and ''x'') can be defined via [[contour integration]], but then depends on choice of path, specifically [[winding number]] – the geometric issue is that the function defines a [[covering space]] with a singularity at&nbsp;0.

=== ''n'' = &minus;1 ===
There is also the exceptional case ''n''&nbsp;=&nbsp;&minus;1, yielding a [[logarithm]] instead of a power of&nbsp;''x:''

:<math>\int_1^a \frac{1}{x}\,dx = \ln a,</math>
:<math>\int \frac{1}{x}\,dx = \ln x + C, \qquad x > 0</math>

(where "ln" means the [[natural logarithm]], i.e. the logarithm to the base ''[[e (mathematical constant)|e]]''&nbsp;=&nbsp;2.71828...).

The improper integral is often extended to negative values of ''x'' via the conventional choice:

:<math>\int \frac{1}{x}\,dx = \ln |x| + C, \qquad x \neq 0.</math>

Note the use of the [[absolute value]] in the indefinite integral; this is to provide a unified form for the integral, and means that the integral of this odd function is an even function, though the logarithm is only defined for positive inputs, and in fact, different constant values of ''C'' can be chosen on either side of 0, since these do not change the derivative. The more general form is thus:<ref>"[http://golem.ph.utexas.edu/category/2012/03/reader_survey_logx_c.html Reader Survey: log|''x''| + ''C'']", Tom Leinster, ''The ''n''-category Café'', March 19, 2012</ref>
:<math>\int\frac{1}{x} \, dx= \begin{cases}
\ln |x| + C^- & x < 0 \\
\ln |x| + C^+ & x > 0
\end{cases}</math>
Over the complex numbers there is not a global antiderivative for 1/''x'', due this function defining a non-trivial [[covering space]]; this form is special to the real numbers.

Note that the definite integral starting from 1 is not defined for negative values of ''a,'' since it passes through a singularity, though since 1/''x'' is an [[odd function]], one can base the definite integral for negative powers at&nbsp;&minus;1. If one is willing to use [[improper integral]]s and compute the [[Cauchy principal value]], one obtains <math>\int_{-c}^c \frac{1}{x}\,dx = 0,</math> which can also be argued by symmetry (since the logarithm is odd), so <math>\int_{-1}^1 \frac{1}{x}\,dx = 0,</math> so it makes no difference if the definite integral is based at 1 or&nbsp;&minus;1. As with the indefinite integral, this is special to the real numbers, and does not extend over the complex numbers.

=== Alternative forms ===
The integral can also be written with indexes shifted, which simplify the result and make the relation to ''n''-dimensional differentiation and the ''n''-cube clearer:
:<math>\int_0^a x^{n-1}\,dx = \tfrac{1}{n} a^n \qquad n \geq 1.</math>
:<math>\int x^{n-1}\,dx = \tfrac{1}{n} x^n + C \qquad n \neq 0.</math>

More generally, these formulae may be given as:
:<math>\int (ax + b)^n dx= \frac{(ax + b)^{n+1}}{a(n + 1)} + C \qquad\mbox{(for } n\neq -1\mbox{)}\,\!</math>
:<math>\int\frac{1}{ax + b} dx= \frac{1}{a}\ln\left|ax + b\right| + C</math>
:More generally:
::<math>\int\frac{1}{ax + b} \, dx= \begin{cases}
\frac{1}{a}\ln\left|ax + b\right| + C^- & x < -b/a \\
\frac{1}{a}\ln\left|ax + b\right| + C^+ & x > -b/a
\end{cases}</math>

== Proof ==
The modern proof is to use an antiderivative: the derivative of ''x''<sup>''n''</sup> is shown to be ''nx''<sup>''n''&minus;1</sup> – for non-negative integers. This is shown from the [[binomial formula]] and the [[definition of the derivative]] – and thus by the [[fundamental theorem of calculus]] the [[antiderivative]] is the integral. This method fails for <math>\int \frac{1}{x}\,dx,</math> as the [[candidate solution|candidate]] antiderivative is <math>\frac{1}{0} \cdot x^0</math>, which is undefined due to division by zero. The [[logarithm]] function, which is the actual antiderivative of 1/''x'', must be introduced and examined separately.

[[Image:BinomialTheorem.png|right|315px|thumb|The derivative <math>(x^n)'=nx^{n-1}</math> can be geometrized as the infinitesimal change in volume of the ''n''-cube, which is the area of ''n'' faces, each of dimension ''n''&nbsp;&minus;&nbsp;1.<br />
Integrating this picture – stacking the faces – geometrizes the fundamental theorem of calculus, yielding a decomposition of the ''n''-cube into ''n'' pyramids, which is a geometric proof of Cavalieri's quadrature formula.]]
For positive integers, this proof can be geometrized:<ref>{{Harv|Barth|2004}}, {{Harv|Carter|Champanerkar|2006}}</ref> if one considers the quantity ''x''<sup>''n''</sup> as the volume of the ''n''-cube (the [[hypercube]] in ''n'' dimensions), then the derivative is the change in the volume as the side length is changed – this is ''x''<sup>''n''&minus;1</sup>, which can be interpreted as the area of ''n'' faces, each of dimension ''n''&nbsp;&minus;&nbsp;1 (fixing one vertex at the origin, these are the ''n'' faces not touching the vertex), corresponding to the cube increasing in size by growing in the direction of these faces – in the 3-dimensional case, adding 3 infinitesimally thin squares, one to each of these faces. Conversely, geometrizing the fundamental theorem of calculus, stacking up these infinitesimal (''n''&nbsp;&minus;&nbsp;1) cubes yields a (hyper)-pyramid, and ''n'' of these pyramids form the ''n''-cube, which yields the formula. Further, there is an ''n''-fold cyclic symmetry of the ''n''-cube around the diagonal cycling these pyramids (for which a pyramid is a [[fundamental domain]]). In the case of the cube (3-cube), this is how the volume of a pyramid was originally rigorously established: the cube has 3-fold symmetry, with fundamental domain a pyramids, dividing the cube into 3 pyramids, corresponding to the fact that the volume of a pyramid is one third of the base times the height. This illustrates geometrically the equivalence between the quadrature of the parabola and the volume of a pyramid, which were computed classically by different means.

Alternative proofs exist – for example, [[Pierre de Fermat|Fermat]] computed the area via an algebraic trick of dividing the domain into certain intervals of unequal length;<ref>See Rickey.</ref> alternatively, one can prove this by recognizing a symmetry of the graph ''y''&nbsp;=&nbsp;''x''<sup>''n''</sup> under inhomogeneous dilation (by ''d'' in the ''x'' direction and ''d''<sup>''n''</sup> in the ''y'' direction, algebraicizing the ''n'' dimensions of the ''y'' direction),<ref>{{Harv|Wildberger|2002}}</ref> or deriving the formula for all integer values by expanding the result for ''n''&nbsp;=&nbsp;&minus;1 and comparing coefficients.<ref>{{Harv|Bradley|2003}}</ref>

== History ==
[[Image:Parabolic Segment.svg|thumb|Archimedes computed the area of parabolic segments in his ''[[The Quadrature of the Parabola]]''.]]
A detailed discussion of the history, with original sources, is given in {{Harv|Laubenbacher|Pengelley|1998|loc=Chapter 3, Analysis: Calculating Areas and Volumes}}; see also [[history of calculus]] and [[Integral#History|history of integration]].

The case of the parabola was proven in antiquity by the ancient Greek mathematician [[Archimedes]] in his ''[[The Quadrature of the Parabola]]'' (3rd century BC), via the [[method of exhaustion]]. Of note is that Archimedes computed the area ''inside'' a parabola – a so-called "parabolic segment" – rather than the area under the graph ''y''&nbsp;=&nbsp;''x''<sup>2</sup>, which is instead the perspective of [[Cartesian geometry]]. These are equivalent computations, but reflect a difference in perspective. The Ancient Greeks, among others, also computed the volume of a [[pyramid (geometry)|pyramid]] or [[cone (geometry)|cone]], which is mathematically equivalent.

In the 11th century, the [[Islamic mathematics|Islamic mathematician]] [[Ibn al-Haytham]] (known as ''Alhazen'' in Europe) computed the integrals of [[cubic polynomial|cubics]] and [[quartic polynomial|quartics]] (degree three and four) via [[mathematical induction]], in his ''[[Book of Optics]]''.<ref name=Katz>Victor J. Katz (1995), "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '''68''' (3): 163–174 [165–9 & 173–4]</ref>

The case of higher integers was computed by Cavalieri for ''n'' up to 9, using his method of indivisibles ([[Cavalieri's principle]]).<ref>{{Harv|Struik|1986|loc=pp. 215–216}}</ref> He interpreted these as higher integrals as computing higher-dimensional volumes, though only informally, as higher-dimensional objects were as yet unfamiliar.<ref>{{Harv|Laubenbacher|Pengelley|1998}} – see [http://www.math.nmsu.edu/~history/book/analysissynopsis.html Informal pedagogical synopsis of the Analysis chapter] for brief form</ref> This method of quadrature was then extended by Italian mathematician [[Evangelista Torricelli]] to other curves such as the [[cycloid]], then the formula was generalized to fractional and negative powers by English mathematician [[John Wallis]], in his ''[[Arithmetica Infinitorum]]'' (1656), which also standardized the notion and notation of rational powers – though Wallis incorrectly interpreted the exceptional case ''n''&nbsp;=&nbsp;&minus;1 (quadrature of the hyperbola) – before finally being put on rigorous ground with the development of [[integral calculus]].

Prior to Wallis's formalization of fractional and negative powers, which allowed ''explicit'' functions <math>y=x^{p/q},</math> these curves were handled ''implicitly,'' via the equations <math>x^p=ky^q</math> and <math>x^py^q=k</math> (''p'' and ''q'' always positive integers) and referred to respectively as '''higher parabolae''' and '''higher hyperbolae''' (or "higher parabolas" and "higher hyperbolas"). [[Pierre de Fermat]] also computed these areas (except for the exceptional case of &minus;1) by an algebraic trick – he computed the quadrature of the higher hyperbolae via dividing the line into equal intervals, and then computed the quadrature of the higher parabolae by using a division into ''unequal'' intervals, presumably by inverting the divisions he used for hyperbolae.<ref>See Rickey reference for discussion and further references.</ref> However, as in the rest of his work, Fermat's techniques were more ad hoc tricks than systematic treatments, and he is not considered to have played a significant part in the subsequent development of calculus.

Of note is that Cavalieri only compared areas to areas and volumes to volumes – these always having ''dimensions,'' while the notion of considering an area as consisting of ''units'' of area (relative to a standard unit), hence being unitless, appears to have originated with Wallis;<ref>Ball, 281</ref><ref>Britannica, 171</ref> Wallis studied fractional and negative powers, and the alternative to treating the computed values as unitless numbers was to interpret fractional and negative dimensions.

The exceptional case of &minus;1 (the standard hyperbola) was first successfully treated by [[Grégoire de Saint-Vincent]] in his ''Opus geometricum quadrature circuli et sectionum coni'' (1647), though a formal treatment had to wait for the development of the [[natural logarithm]], which was accomplished by [[Nicholas Mercator]] in his ''Logarithmotechnia'' (1668).

== References ==
{{reflist}}

=== History ===
{{refbegin}}
* Cavalieri, ''Geometria indivisibilibus (continuorum nova quadam ratione promota)'' (Geometry, exposed in a new manner with the aid of indivisibles of the continuous), 1635.
* Cavalieri, ''Exercitationes Geometricae Sex'' ("Six Geometrical Exercises"), 1647
** in [[Dirk Jan Struik]], editor, ''A source book in mathematics, 1200–1800'' (Princeton University Press, Princeton, New Jersey, 1986). {{ISBN|0-691-08404-1}}, {{ISBN|0-691-02397-2}} (pbk).
* ''Mathematical expeditions: chronicles by the explorers,'' Reinhard Laubenbacher, David Pengelley, 1998, Section 3.4: "Cavalieri Calculates Areas of Higher Parabolas", [https://books.google.com/books?id=ubFg7DI83JoC&pg=PA125 pp. 123–127/128]
* ''A short account of the history of mathematics,'' Walter William Rouse Ball, "Cavalieri", [https://books.google.com/books?id=_sT_psl3uYkC&pg=PA278 p. 278–281]
* "[http://eom.springer.de/i/i050950.htm Infinitesimal calculus]", ''Encyclopaedia of Mathematics''
* ''The Britannica Guide to Analysis and Calculus,'' by Educational Britannica Educational, [https://books.google.com/books?id=ML5Uuo16D58C&pg=PA171 p. 171] – discusses Wallace primarily
{{refend}}

=== Proofs ===
{{refbegin}}
* {{cite journal
|last1=Wildberger
|first1=N. J.
|title=A new proof of Cavalieri's quadrature formula
|journal=The American Mathematical Monthly
|volume=109
|issue=9
|pages=843–845
|year=2002
|doi=10.2307/3072373
|jstor=3072373
}}
* {{Cite journal
| last = Bradley
| first = David M.
| title = Remark on Cavalieri's quadrature formula
| arxiv = math/0505059
| journal = The American Mathematical Monthly
| volume = 110| issue = 5| page = 437
|date=May 2003
| bibcode = 2005math......5059B
| postscript =, appeared in print at end of [https://www.jstor.org/pss/3647831 Zeros of the Alternating Zeta Function on the Line R(S) = 1]
}}
* {{cite journal
|last1 = Barth
|first1 = N. R.
|title = Computing Cavalieri's quadrature formula by a symmetry of the n-cube
|journal = The American Mathematical Monthly
|volume = 111
|issue = 9
|pages = 811–813
|year = 2004
|doi = 10.2307/4145193
|jstor = 4145193
}}
* {{Cite arxiv
| last1 = Carter
| first1 = J. Scott
| last2 = Champanerkar
| first2 = Abhijit
| title = A geometric method to compute some elementary integrals
| eprint = math/0608722
| year = 2006
}}
* Malik, M.A. (1984) "A Note on Cavalieri Integration", [[Mathematics Magazine]] 57(3): 154–6 {{doi|10.2307/2689662}}
* [https://web.archive.org/web/20110720102654/http://www.math.usma.edu/people/rickey/ V. Frederick Rickey] (2011) [https://web.archive.org/web/20110720101905/http://www.math.usma.edu/people/rickey/hm/CalcNotes/Fermat-Integration.pdf Fermat's Integration of Powers]", in ''[https://web.archive.org/web/20110720102613/http://www.math.usma.edu/people/rickey/hm/CalcNotes/default.htm Historical Notes for Calculus Teachers]'' 
{{refend}}

== External links ==
* {{mathworld |title=Cavalieri's Quadrature Formula |id=CavalierisQuadratureFormula }}
* [http://researchspace.csir.co.za/dspace/bitstream/10204/5267/1/Grobler5_2011.pdf Cavalieri Integration]
* [https://books.google.com/books?id=XmRsZhJZGhEC&lpg=PA214&hl=fr&pg=PA214#v=onepage&q&f=false D. J. Struik, ''A Source Book in Mathematics, 1200-1800'', p. 214]

[[Category:Integrals]]
[[Category:Polynomials]]