The principles of integration apply in each case.
Usually single integrals can be definite (with limits) or indefinite (constant of integration is required).
Multiple integrals usually have limits for the innermost integrals so that the outermost integral can be evaluated as a single integral, definite or indefinite. If the innermost integrals don’t have limits, constants of integration have to be introduced.
In multiple integrals, variables are regarded as constants when the variable is not the one explicitly indicated in the infinitesimal (dx, dy, etc.). In single integrals there can be only one variable and its associated infinitesimal if the integral is to be evaluated.
To be evaluated each integral in a multiple integral must be associated with one and only one variable, and there must be as many integral as there are variables. The innermost integral is evaluated first with respect to the leftmost infinitesimal. When this integral has been evaluated, the next innermost integral is evaluated with respect to the infinitesimal on the right of the one just used. And so on until the outermost integral becomes the only one left to evaluate.
