Multiple integral (${\int }_{D}f(x_1,\dots ,x_n)d{x}_1\cdots d{x}_n$)

A multiple integral accumulates a function over a region with several coordinates. The Fubini theorem and Tonelli theorem give conditions under which it equals successive one-dimensional integrals. A change of variables formula expresses it in new coordinates using the absolute Jacobian determinant.

A triple integral is a multiple integral over a three-dimensional region. For a mass density $\rho$, ${\iiint }_D\rho dV$ gives the total mass, while ${\iiint }_{D}1dV$ gives its volume. Iterated one-dimensional integration is justified by Fubini's theorem under absolute integrability, and a change of variables includes the absolute Jacobian determinant in the volume element.

Under a smooth locally invertible coordinate map $\Phi$ into Euclidean space, the positive volume element in a multiple integral is multiplied by the absolute Jacobian determinant. In three-dimensional spherical coordinates, $dV=r^2\sin \theta drd\theta d\phi$; in cylindrical coordinates, $dV=rdrd\theta dz$. The absolute value distinguishes positive integration measure from a signed orientation convention.

An iterated integral performs successive one-variable integrals. In the displayed expression, $y$ is held fixed during the inner integration over $x$, and the result is then integrated over $y$. The inner bounds may depend on that fixed outer variable. By contrast, an inner bound depending on an already integrated variable does not specify such a nested operation without further interpretation. For example, the triangle $0\le y\le 1$, $2y\le x\le 2$ gives ${\int }_0^1{\int }_{2y}^{2}f(x,y)dxdy$, or, in the other order, ${\int }_0^2{\int }_0^{x/2}f(x,y)dydx$. Fubini's theorem and Tonelli's theorem supply conditions under which these iterated expressions equal the same multiple integral.

The sequence in which variables are integrated in an iterated integral. Reversing the order changes the slicing of the domain of integration, often requiring more than one integral. Fubini's theorem permits interchange for absolutely integrable functions, while Tonelli's theorem permits nonnegative measurable integrands even when the integral is infinite. Conditional convergence requires additional care; geometric reversal alone is not a general justification for interchanging improper integrals.

If a region is invariant under one coordinate reflection and an integrable function changes sign under that reflection, its multiple integral is zero. Apply the change of variables formula to the reflection, whose absolute Jacobian determinant is one: the integral equals its negative. This applies to odd coordinate powers in symmetric volumes and to the off-diagonal entries of an inertia tensor when the mass density shares the coordinate-reflection symmetries.