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IEVref: | 102-04-33 | ID: | |

Language: | en | Status: Standard | |

Term: | area | ||

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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 Note 2 to entry: For a surface defined by $r=f(u\text{\hspace{0.17em}},\text{\hspace{0.17em}}v)$ where $(u,v)\in \text{U}\subseteq {R}^{\text{2}}$, the area is $\underset{\text{U}}{\iint}\left|\frac{\partial f}{\partial u}\times \frac{\partial f}{\partial v}\right|\mathrm{d}u\mathrm{d}v$. Note 3 to entry: For a surface defined by the equation 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 | ||

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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 | ||

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- 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 $\underset{a}{\overset{b}{\int}}f(x)\mathrm{d}x$.

Note 2 to entry: For a surface defined by $r=f(u\text{\hspace{0.17em}},\text{\hspace{0.17em}}v)$ where $(u,v)\in \text{U}\subseteq {R}^{\text{2}}$, the area is $\underset{\text{U}}{\iint}\left|\frac{\partial f}{\partial u}\times \frac{\partial f}{\partial v}\right|\mathrm{d}u\mathrm{d}v$.

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.