Editing Riemann sum (section)

== Higher dimensions ==
The basic idea behind a Riemann sum is to "break-up" the domain via a partition into pieces, multiply the "size" of each piece by some value the function takes on that piece, and sum all these products. This can be generalized to allow Riemann sums for functions over domains of more than one dimension.

While intuitively, the process of partitioning the domain is easy to grasp, the technical details of how the domain may be partitioned get much more complicated than the one dimensional case and involves aspects of the geometrical shape of the domain.<ref>{{cite book |last=Swokowski |first=Earl W. |year=1979 |title=Calculus with Analytic Geometry |url=https://archive.org/details/studentsupplemen00bron |url-access=registration |edition=Second |publisher=Prindle, Weber & Schmidt |location=Boston, MA |isbn=0-87150-268-2 |pages=821–822}}</ref>

=== Two dimensions ===
In two dimensions, the domain <math>A</math> may be divided into a number of two-dimensional cells <math>A_i</math> such that <math display="inline">A = \bigcup_i A_i</math>. Each cell then can be interpreted as having an "area" denoted by <math>\Delta A_i</math>.<ref>{{cite book |last1=Ostebee |first1=Arnold |last2=Zorn |first2=Paul |year=2002 |title=Calculus from Graphical, Numerical, and Symbolic Points of View |edition=Second |page=M-34 |quote=We chop the plane region ''R'' into ''m'' smaller regions ''R''<sub>1</sub>, ''R''<sub>2</sub>, ''R''<sub>3</sub>, ..., ''R''<sub>''m''</sub>, perhaps of different sizes and shapes. The 'size' of a subregion ''R''<sub>''i''</sub> is now taken to be its ''area'', denoted by Δ''A''<sub>''i''</sub>.}}</ref> The two-dimensional Riemann sum is
<math display="block">S = \sum_{i = 1}^n f(x_i^*, y_i^*)\, \Delta A_i,</math>
where <math>(x_i^*, y_i^*) \in A_i</math>.

=== Three dimensions ===
In three dimensions, the domain <math>V</math> is partitioned into a number of three-dimensional cells <math>V_i</math> such that <math display="inline">V = \bigcup_i V_i</math>. Each cell then can be interpreted as having a "volume" denoted by <math>\Delta V_i</math>. The three-dimensional Riemann sum is<ref>{{cite book |last=Swokowski |first=Earl W. |year=1979 |title=Calculus with Analytic Geometry |url=https://archive.org/details/studentsupplemen00bron |url-access=registration |edition=Second |publisher=Prindle, Weber & Schmidt |location=Boston, MA |isbn=0-87150-268-2 |pages=857–858}}</ref>
<math display="block">S = \sum_{i = 1}^n f(x_i^*, y_i^*, z_i^*)\, \Delta V_i,</math>
where <math>(x_i^*, y_i^*, z_i^*) \in V_i</math>.

=== Arbitrary number of dimensions ===
Higher dimensional Riemann sums follow a similar pattern. An ''n''-dimensional Riemann sum is
<math display="block">S = \sum_i f(P_i^*)\, \Delta V_i,</math>
where <math>P_i^* \in V_i</math>, that is, it is a point in the ''n''-dimensional cell <math>V_i</math> with ''n''-dimensional volume <math>\Delta V_i</math>.

=== Generalization ===
In high generality, Riemann sums can be written
<math display="block">S = \sum_i f(P_i^*) \mu(V_i),</math>
where <math>P_i^*</math> stands for any arbitrary point contained in the set <math>V_i</math> and <math>\mu</math> is a [[Measure (mathematics)|measure]] on the underlying set. Roughly speaking, a measure is a function that gives a "size" of a set, in this case the size of the set <math>V_i</math>; in one dimension this can often be interpreted as a length, in two dimensions as an area, in three dimensions as a volume, and so on.