Calculus How To

Cubature: Definition, Examples

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In a general sense of the word, “cubature” is the process for finding the volume of a solid. In calculus, cubature is defined as a numerical computation method for multiple integrals, including those on higher dimensions. For example, on three-dimensional solids or four-dimensional hypercubes, where the limits of integration are vectors. This is in comparison to quadrature, which is numerical computation for a single integral [1].


Triple integral for volume of a cube.

The basic idea is similar to Riemann sums: The figure is divided into a series of rectangles and finding the limit as these rectangles become thinner. For shapes in higher dimensions (say, over 7), Monte Carlo or similar methods are used [2].

Where to Find Cubature Formula

As you can probably imagine, the number of ways to perform cubature is vast. There are so many in fact, that entire encyclopedias are dedicated to them. Most academic papers on multiple numerical integration refer to one of two books:

  • A.H. Stroud. Approximate Calculation of Multiple Integrals, Prentice–Hall, Englewood Cliffs, NJ (1971). This edition is quite comprehensive. Unfortunately, a copy will run you about $500.
  • I.P. Mysovskikh. Interpolatory Cubature Formulas, Izdat. Nauka, Moscow, Leningrad (1981). Also a comprehensive book, but the problem is it’s written in the Russian language.

The website Encyclopaedia of Cubature Formulas, developed in Belgium at the Department of Computer Science Katholieke Universiteit Leuven, contains a wealth of cubature tables. For example, you can find formulas here for the cube (including n-cube) and sphere (including n-sphere). Unfortunately, it’s an older site and not very easy to navigate or access (certain parts of the site tell you to login, with no further instructions).

A better solution is to use the formulas found in some statistical software. For example, this R package is for adaptive multivariate integration over hypercubes.


[1] Krommer, A. R. and Ueberhuber, C. W. “Construction of Cubature Formulas.” §6.1 in Computational Integration. Philadelphia, PA: SIAM, pp. 155-165, 1998.
[2] Cubature (Multi-dimensional integration). Retrieved April 8, 2021 from:

Stephanie Glen. "Cubature: Definition, Examples" From Calculus for the rest of us!

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