## The Mean Value Theorem for Multiple Integrals

As in the single variable case, multiple integrals satisfy a mean value property.

First of all, we should note that, like one-dimensional Riemann-Stieltjes integrals with increasing integrators, integration preserves order. That is, if and are both integrable over a Jordan-measurable set , and if at each point , then we have

This is a simple consequence of the definition of a multiple integral as the limit of Riemann sums, since every Riemann sum for will be smaller than the corresponding sum for .

Now if and are integrable on and for every , then we set and — the infimum and supremum of the values attained by on . I assert that there is some in the interval so that

In particular, we can set and find

giving bounds on the integral in terms of the Jordan content of . Incidentally, here is serving a similar role to the integrator in the integral mean value theorem for Riemann-Stieltjes integrals.

Okay, so since we have for every . Since integration preserves order, this yields

If the integral of is zero, then our result automatically holds for any value of . Otherwise we can divide through by this integral and set

which will be between and .

One particularly useful case is when has Jordan content zero. In this case, we find that any integral over is itself automatically zero.

[…] in for integrals are Jordan measurable, and their boundaries have zero Jordan content, so we know changing things along these boundaries in an integral will make no […]

Pingback by Integrals are Additive Over Regions « The Unapologetic Mathematician | December 30, 2009 |

[…] infinitesimal pieces of -dimensional volume. Now, with the change of variables formula and the mean value theorem in hand, we can pull out a macroscopic […]

Pingback by The Geometric Interpretation of the Jacobian Determinant « The Unapologetic Mathematician | January 8, 2010 |

[…] analogue of the integral mean value theorem that holds not just for single integrals, not just for multiple integrals, but for integrals over any measure […]

Pingback by The Integral Mean Value Theorem « The Unapologetic Mathematician | June 14, 2010 |