  If we wish to integrate a function $h|\Omega\subset\Re^{n}\to\Re$ along a
single $T$-level surface of a function $\psi |\Omega\subset\Re^{n}\to\Re$, then
a number of different methods for extracting finite elements appropriate to the
dimension of the level surface may be employed to obtain an explicit
representation over which the integration may be performed using standard
numerical quadrature techniques along each element. However, when the goal is
to compute an entire continuous family $m(T)$ of integrals over all the
$T$-level surfaces of $\psi$, then this method of explicit level set extraction
is no longer practical. We introduce a novel method to perform this type of
numerical integration efficiently by making use of the coarea formula. We
present the technique for discretization of the coarea formula and present the
algorithms to compute the integrals over families of T-level surfaces. While
validation of our method in the special case of a single level surface
demonstrates accuracies close to more explicit isosurface integration methods,
we show a sizable boost in computational efficiency in the case of multiple
T-level surfaces, where our coupled integration algorithms significantly
outperform sequential one-at-a-time application of explicit methods.
