Cavalieri's quadrature formula

In calculus, Cavalieri's quadrature formula, named for 17th-century Italian mathematician Bonaventura Cavalieri, is the integral

${\displaystyle \int _{0}^{a}x^{n}\,dx={\tfrac {1}{n+1}}\,a^{n+1}\qquad n\geq 0,}$

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

${\displaystyle \int x^{n}\,dx={\tfrac {1}{n+1}}\,x^{n+1}+C\qquad n\neq -1.}$

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.