Given a surface S or volume V and an integrand f(x, y, z), defined and reasonably well behaved in S or V, we can form the analogue of Riemann sums in this context.
For volume: break V up into small pieces and sum a value of f in each piece times  its volume; take the limit as the volumes of all pieces goes to zero in every possible way. If it exists this limit is the multiple integral of f over V. The limit can be shown to exist when f is continuous and V is compact (bounded and closed). It often exists otherwise as well.

Exercise

Iterated integrals
Any product which contains two or more integrals in it can be considered an iterated integral.

Examples

Notation