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The definite integral \(\int _a^b f(x) \mathrm {d} x\) provides the area enclosed between \([a,b] \subseteq {\mathbb R}\) and the graph of f. This concept can be easily generalised: In an area integral \(\int _D f(x_1 ,\dots , x_n) \mathrm {d} x_1 \cdots \mathrm {d} x_n\) , the volume is determined that is enclosed between the area \(D \subseteq {\mathbb R}^n\) and the graph of f. If D is a subset of \({\mathbb R}^2\) , then this is a (three-dimensional) volume.

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Area Integrals

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  • 发表日期：

    2026年

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  • Christian Karpfinger

摘要

The definite integral \(\int _a^b f(x) \mathrm {d} x\) provides the area enclosed between \([a,b] \subseteq {\mathbb R}\) and the graph of f. This concept can be easily generalised: In an area integral \(\int _D f(x_1 ,\dots , x_n) \mathrm {d} x_1 \cdots \mathrm {d} x_n\) , the volume is determined that is enclosed between the area \(D \subseteq {\mathbb R}^n\) and the graph of f. If D is a subset of \({\mathbb R}^2\) , then this is a (three-dimensional) volume.