Name: double integrals on curved boundaries
Uploaded: 2026-06-08T09:38:44.708Z
Duration: 1 min 18 s
Channel: JoVE

考虑一个形状不规则的区域 D，它是一处池塘的一部分，被包围在一个矩形公园内，该公园区域记为 R。由于池塘边界呈弯曲且不规则的形状，直接在区域 D 上进行积分可能非常复杂。为了简化积分，定义了一个新的函数 g，它在整个矩形区域上与原函数 f 在区域 D 内的取值一致，而在池塘边界外的所有位置均等于零。在实际应用中，通过使用积分变量的上下限，将二重积分作为累次积分进行计算。如果该区域在水平轴上介于两条恒定的垂直线之间，并且从下边界曲线延伸到上边界曲线，则被归类为Ⅰ型区域。或者，如果该区域在垂直轴上介于两条恒定的水平线之间，且从左侧边界曲线移动到右侧边界曲线，则被归类为II型区域。通过将这些曲线边界描述为可变函数，可以无缝计算任意一般区域上的二重积分。

一般区域上的二重积分

二重积分常用于测量分布在二维区域上的物理量，例如湖泊上的降雨量、金属板上的热量分布或陆地上的种群密度。在许多实际情况下，感兴趣的区域并不具有直线边界，无法方便地用矩形来描述。相反，该区域可能具有曲线或不规则的边缘。为了在这些区域上计算积分，通常将该区域嵌入到一个更大的矩形区域内，以便于应用积分方法。将二重积分推广到不规则区域设 D 表示矩形区域 R 内的一个不规则区域。假设 f(x,y) 描述了分布在 D 上的某种物理量，例如湖泊上方的降雨强度。现定义一个新函数 F(x,y)，其定义域为整个矩形区域 R，定义方式如下：\begin{equation*}F(x,y)=\begin{cases}f(x,y)，且 (x,y)∈D \\0，且 (x,y) 属于 R 但不属于 D\end{cases}\end{equation*}这种构造允许在矩形 R 上进行积分，同时确保只有 D 内部的值对结果有贡献。由于在不规则区域外 F(x,y) = 0，矩形的周围部分对积分无贡献。因此，一般区域 D 上的二重积分可表示为\begin{equation*}\iint_D f(x,y)\, dA = \iint_R F(x,y)\, dA\end{equation*}I型区域与II型区域不规则区域通常分为两种类型。I型区域为垂直边界区域，可表示为\begin{equation*}D={(x,y)∣a≤x≤b, g₁(x)≤y≤g₂(x)}\end{equation*}II型区域在水平方向上有界，其表示形式为\begin{equation*}D={(x,y)∣c≤y≤d, h₁(y)≤x≤h₂(y)}\end{equation*}这些描述将曲面区域上的积分转化为具有变量限的累次积分，从而实现对分布在不规则区域上的量进行系统性求解。

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Multiple Integrals and Applications

多重积分及其应用

KEY TERMS AND CONCEPTS

关键术语与概念

Double Integrals Over General Regions

一般区域上的双积分

double integrals on curved boundaries

Double integrals can measure quantities spread across a two-dimensional region, such as rainfall over a lake, heat on a metal plate, or population density over land. In many real situations, the region does not have straight sides. It may have curved or irregular edges instead. That makes the region harder to describe as a simple rectangle.

To handle an irregular region D, it is placed inside a larger rectangular region R. A function f(x,y) gives the quantity on D, such as rainfall intensity over a lake. A new function F(x,y) is then defined on all of R. It equals f(x,y) inside D and 0 everywhere in R that lies outside D.

This setup lets the integral be taken over the easier rectangular region R. The outside parts of the rectangle contribute nothing because F(x,y)=0 there. The double integral over the general region D is written as the double integral over R of F(x,y) dA.

Irregular regions are often described as Type I or Type II regions. A Type I region is bounded vertically. It is written with x running from a to b, while y runs from g1(x) to g2(x).

A Type II region is bounded horizontally. It is written with y running from c to d, while x runs from h1(y) to h2(y). These forms turn double integrals over curved regions into iterated integrals with variable limits. That makes the calculation of quantities over irregular domains more systematic.

Double Integrals on Curved Boundaries

introduction to double integrals

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double integrals over general regions

Double integrals can measure quantities spread across a two-dimensional region, such as rainfall over a lake, heat on a metal plate, or population density over land. In many real situations, the region does not have straight sides. It may have curved or irregular edges instead. That makes the region harder to describe as a simple rectangle. To handle an irregular region D, it is placed inside a larger rectangular region R. A function f(x,y) gives the quantity on D, such as rainfall intensity...

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triple integrals in rectangular coordinates

triple-integrals-in-rectangular-coordinates

Triple Integrals for Volume and Mass

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triple integrals over general regions

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Setting Triple Integral Bounds for Easier Evaluation

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triple integrals in cylindrical coordinates

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triple integrals in spherical coordinates

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Finding Sphere Volume with Spherical Wedges

Triple integrals in spherical coordinates are a useful way to find volumes with central symmetry, especially spheres. Spherical coordinates describe a point with three variables: ρ, θ, and φ. Here, ρ is the distance from the origin. θ is the angle in the xy-plane measured from the positive x-axis, and φ is the angle measured downward from the positive z-axis. To find the volume of a sphere, the solid can be split into many small spherical wedges. Each wedge is defined by small changes in ρ, θ,...

change of variables in multiple integrals

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Using the Jacobian to Simplify Integrals

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real-life applications of multiple integrals

real-life-applications-of-multiple-integrals

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key terms and concepts

Learn how double integrals handle irregular regions by using a larger rectangle, a zero-outside function, and Type I and Type II region bounds.

Double Integrals on Curved Boundaries | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Double Integrals on Curved Boundaries

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