Cavalieri's quadrature formula
In calculus, Cavalieri's quadrature formula, named for 17th-century Italian mathematician Bonaventura Cavalieri, is the integral
and generalizations thereof. This is the definite integral form; the indefinite integral form is:
There are additional forms, listed below. Together with the linearity of the integral, this formula allows one to compute the integrals of all polynomials.
The term "quadrature" is a traditional term for area; the integral is geometrically interpreted as the area under the curve y = xn. Traditionally important cases are y = x2, the quadrature of the parabola, known in antiquity, and y = 1/x, the quadrature of the hyperbola, whose value is a logarithm.
For negative values of n (negative powers of x), there is a singularity at x = 0, and thus the definite integral is based at 1, rather than 0, yielding:
Further, for negative fractional (non-integer) values of n, the power xn is not well-defined, hence the indefinite integral is only defined for positive x. However for n a negative integer the power xn 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 −x, and basing the negative definite integral at −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 0.
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