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
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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.