So a chain is actually pretty simple; it’s just a formal linear combination of singular -cubes. That is, for each we build the free abelian group generated by the singular -cubes in .
If we have a formal sum — the are all singular -cubes and the are all integers — then we define integrals over the chain by linearity:
And that’s all there is to it; just cover the -dimensional region you’re interested in with singular -cubes. If there are some overlaps, those areas will get counted twice, so you’ll have to cover them with their own singular -cubes with negative multipliers to cancel them out. Take all the integrals — by translating each one back to the standard -cube — and add (or subtract) them up to get the result!