# multiple integrals

(3.4 hours to learn)

## Summary

A multiple integral generalizes integration to functions of n variables and produces a general (n-1)-dimensional volume. For instance, n=2 corresponds to an area. Multiple integrals occur frequently in probability theory and machine learning when examining marginal densities.

## Context

This concept has the prerequisites:

## Core resources (read/watch one of the following)

## -Free-

→ MIT Open Courseware: Multivariable Caclulus (2010)

Video lectures for MIT's introductory multivariable calculus class.

→ Khan Academy: Calculus

## -Paid-

→ Multivariable Calculus

An introductory multivariable calculus textbook.

- Section 14.1, "Double integrals," pages 940-945
- Section 14.2, "Double integrals over more general regions," pages 947-952
- Section 14.3, "Area and volume by double integration," pages 954-958
- and Section 14.6, "Triple integrals," pages 979-985

## Supplemental resources (the following are optional, but you may find them useful)

## -Free-

→ Wolfram MathWorld

## -Paid-

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

- Section 7.1, "Multiple integrals," pages 267-273
- and Section 7.2, "Iterated integrals and Fubini's theorem," pages 276-284

Other notes:

- This presents the Riemann integral, a precise formulation of the intuitive notion of multiple integrals, but which unfortunately has some awkward notation.

## See also

- Some uses of multiple integrals:
- Computing the probability that a multivariate continuous random variable takes a value in some set

- Some tricks for evaluating multiple integrals:
- changing the order of integration
- change of variables
- polar coordinates
- spherical coordinates
- cylindrical coordinates