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.