The integral of a function of three variables is similar to the integral of a function of two variables. In this case, the term: “mesh” refers to a collection of little boxes which covers a given region in R.
Definition 14.3.1 Let R be a bounded region in the ℝ3 and let f be a bounded function defined on R. We say f is Riemannn integrable if there exists a number, denoted by ∫ RfdV and called the Riemannn integral such that if ε > 0 is given, then whenever one imposes a sufficiently fine mesh enclosing R and considers the finitely many boxes which intersect R, numbered as
Of course one can continue generalizing to higher dimensions by analogy. By exactly similar reasoning to the case of integrals of functions of two variables, we can consider iterated integrals as a tool for finding the Riemannn integral of a function of three or more variables.