Cavalieri's quadrature formula
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 0.
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 −1.
If one is willing to use improper integrals and compute the Cauchy principal value, one obtains
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: More generally, these formulae may be given as: The modern proof is to use an antiderivative: the derivative of xn is shown to be nxn−1 – 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.
The logarithm function, which is the actual antiderivative of 1/x, must be introduced and examined separately.
For positive integers, this proof can be geometrized:[2] if one considers the quantity xn 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 xn−1, which can be interpreted as the area of n faces, each of dimension n − 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.
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, Fermat computed the area via an algebraic trick of dividing the domain into certain intervals of unequal length;[3] alternatively, one can prove this by recognizing a symmetry of the graph y = xn under inhomogeneous dilation (by d in the x direction and dn in the y direction, algebraicizing the n dimensions of the y direction),[4] or deriving the formula for all integer values by expanding the result for n = −1 and comparing coefficients.
The Ancient Greeks, among others, also computed the volume of a pyramid or cone, which is mathematically equivalent.
In the 11th century, the Islamic mathematician Ibn al-Haytham (known as Alhazen in Europe) computed the integrals of cubics and quartics (degree three and four) via mathematical induction, in his Book of Optics.
[8] 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 = −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
Pierre de Fermat also computed these areas (except for the exceptional case of −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.
[9] 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;[10][11] 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 −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).
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.
