“multiple integral”的概念、定义、翻译、参考文献-科学参考

multiple integral

Mathematics

An integral involving two or more integrations with respect to different variables, where the integrations are carried out sequentially and variables other than the one being integrated are treated as constants. A double integral is the special case where two integrations are involved. Multiple integrals can be definite or indefinite. For example,

$\begin{array}{l} \int_{0}^{3} \int_{0}^{2} (x + 3 y + 5) d x d y \\ = \int_{0}^{3} {[\frac{x^{2}}{2} + 3 x y + 5 x]}_{0}^{2} d y \\ = \int_{0}^{3} (6 y + 7) d y = {[3 y^{2} + 7 y]}_{0}^{3} = 48. \end{array}$

This double integral represents the volume under the plane z = x + 3y + 5 that lies above the rectangle 0 ≤ x ≤ 2, 0 ≤ y ≤ 3.

单词	multiple integral
释义	multiple integral Mathematics An integral involving two or more integrations with respect to different variables, where the integrations are carried out sequentially and variables other than the one being integrated are treated as constants. A double integral is the special case where two integrations are involved. Multiple integrals can be definite or indefinite. For example, $\begin{array}{l} \int_{0}^{3} \int_{0}^{2} (x + 3 y + 5) d x d y \\ = \int_{0}^{3} {[\frac{x^{2}}{2} + 3 x y + 5 x]}_{0}^{2} d y \\ = \int_{0}^{3} (6 y + 7) d y = {[3 y^{2} + 7 y]}_{0}^{3} = 48. \end{array}$ This double integral represents the volume under the plane z = x + 3y + 5 that lies above the rectangle 0 ≤ x ≤ 2, 0 ≤ y ≤ 3.
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