IEVref: 102-04-33 ID: Language: en Status: Standard Term: area Synonym1: Synonym2: Synonym3: Symbol: 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 integralNote 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 $\underset{a}{\overset{b}{\int }}f\left(x\right)\mathrm{d}x$. Note 2 to entry: For a surface defined by $r=f\left(u\text{\hspace{0.17em}},\text{\hspace{0.17em}}v\right)$ where $\left(u,v\right)\in \text{U}\subseteq {R}^{\text{2}}$, the area is $\underset{\text{U}}{\iint }|\frac{\partial f}{\partial u}×\frac{\partial f}{\partial v}|\mathrm{d}u\mathrm{d}v$. Note 3 to entry: For a surface defined by the equation z = f(x, y), the area is $\underset{\text{S}}{\iint }\sqrt{1+{\left(\frac{\partial f}{\partial x}\right)}^{2}+{\left(\frac{\partial f}{\partial y}\right)}^{2}}\mathrm{d}x\mathrm{d}y$. Note 4 to entry: In the usual geometrical space, the area of a surface is a quantity of the dimension length squared. Publication date: 2008-08 Source: Replaces: Internal notes: 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO 2021-03-15: Formula in Note 2 to entry corrected to replace the second ∂u by ∂v. JGO 2021-04-09: Correction of error in the original standard: the multiplication sign between the vectors shall be a cross and the second multiplication dot after the bars shall be omitted. JGO CO remarks: TC/SC remarks: VT remarks: Domain1: Domain2: Domain3: Domain4: Domain5: