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 ![[m,M]](http://s0.wp.com/latex.php?latex=%5Bm%2CM%5D&bg=e6e6e6&fg=333333&s=0 "[m,M]")
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.
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December 29, 2009 -
Posted by John Armstrong |
Analysis, Calculus
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