Riemann sum
Three dimensions

In three dimensions, the domain V {\displaystyle V} is partitioned into a number of three-dimensional cells V i {\displaystyle V_{i}} such that V = ⋃ i V i {\textstyle V=\bigcup _{i}V_{i}}. Each cell then can be interpreted as having a "volume" denoted by Δ V i {\displaystyle \Delta V_{i}}. The three-dimensional Riemann sum is S = ∑ i = 1 n f (x i ∗, y i ∗, z i ∗) Δ V i, {\displaystyle S=\sum _{i=1}^{n}f(x_{i}^{*},y_{i}^{*},z_{i}^{*})\,\Delta V_{i},} where (x i ∗, y i ∗, z i ∗) ∈ V i {\displaystyle (x_{i}^{*},y_{i}^{*},z_{i}^{*})\in V_{i}}.
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