  Much of the vast literature on the integral during the last two centuries
concerns extending the class of integrable functions. In contrast, our
viewpoint is akin to that taken by Hassler Whitney [{\it Geometric integration
theory}, Princeton Univ. Press, Princeton, NJ, 1957] and by geometric measure
theorists because we extend the class of integrable {\it domains}. Let $\omega$
be an $n$-form defined on $\Bbb R^m$. We show that if $\omega$ is sufficiently
smooth, it may be integrated over sufficiently controlled, but nonsmooth,
domains $\gamma$. The smoother is $ \omega$, the rougher may be $\gamma$.
Allowable domains include a large class of nonsmooth chains and topological
$n$-manifolds immersed in $\Bbb R^m$. We show that our integral extends the
Lebesgue integral and satisfies a generalized Stokes' theorem.
