Definition: | unique positive value, if it exists, associated with a subset of a surface in the three-dimensional Euclidean space, with the following properties: - for a rectangle, the value is the product of the two side lengths,
- for a disjoint union of subsets, the value is the sum of the values associated with these subsets,
- for more complicated subsets, the value can be approximated by sums and given by an integral
Note 1 to entry: For the portion of plane limited by the straight lines x = a, x = b, y = 0 and the arc of curve y = f(x) with a < b and f(x) ≥ 0, the area is b∫af(x)dx. Note 2 to entry: For a surface defined by r=f(u , v) where (u,v)∈U⊆R2, the area is ∬. Note 3 to entry: For a surface defined by the equation z = f(x, y), the area is . Note 4 to entry: In the usual geometrical space, the area of a surface is a quantity of the dimension length squared.
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