Name: evaluating triple integrals by region
Uploaded: 2026-06-08T09:38:52.664Z
Duration: 1 min 28 s
Channel: JoVE

一般区域上的三重积分描述了在三维空间中有界立体上的累积量。区域 E 可能具有弯曲或不规则的边界，因此需要一种系统的方法来确定积分限。为了定义在 E 上的积分，将该立体置于一个更大的矩形盒子内。定义一个新函数，在 E 内部与原函数相同，在立体外部为零。这样就利用外围的盒子定义了在 E 上的三重积分。但在实际计算时，可通过垂直切片来描述该立体。一个实际的例子是估算地下矿床中矿石的含量，该矿床的顶面和底面由两个函数给出。该沉积物在水平面上的投影形成一个基底区域 D。D 中的每一点均对应穿过该立体的一条垂直线段，从下表面延伸至上表面。然后使用垂直切片法将三重积分作为累次积分进行计算。在内层积分中，沿垂直线段对数值进行累加，然后在基区域 D 上对这些部分和进行积分。因此，三重积分给出了该固体内部矿石的总量。

一般区域上的三重积分

对一般有界区域的三重积分将平面区域上的二重积分概念推广到三维立体区域。空间中的一个立体区域 E 通常被包含在一个矩形盒子 B 内，通过对函数 f(x, y, z) 定义扩展函数 F，使其在 E 上与 f 一致而在立体区域外为零，从而在该区域上对连续函数进行积分。因此，三重积分可表示为\begin{equation*}\iiint_E f(x,y,z) dV \end{equation*}积分的存在性要求函数 f 连续，且区域 E 的边界足够光滑。与二重积分类似，三重积分的计算在很大程度上依赖于区域的几何描述。一类常见的立体是I型区域，其中变量z被限制在两个连续曲面之间：\begin{equation*}E=\{(x,y,z)\mid (x,y)\in D; u_1(x,y)\le z\le u_2(x,y)\}\end{equation*}此处，D 表示该立体在 xy-平面上的投影。z = u1(x, y) 给出该立体的下边界，而上边界由 z = u2(x, y) 描述。在这种情况下，三重积分可重写为累次积分：\begin{equation*}\iiint_E f(x,y,z)\, dV=\iint_D\left[\int_{u_1(x,y)}^{u_2(x,y)}f(x,y,z)\, dz\right] dA\end{equation*}如果投影 D 是一个 I 型平面区域，\begin{equation*}a\le x\le b,\qquad g_1(x)\le y\le g_2(x)\end{equation*}积分变为\begin{equation*}\int_a^b\int_{g_1(x)}^{g_2(x)}\int_{u_1(x,y)}^{u_2(x,y)}f(x,y,z) dz\, dy\, dx\end{equation*}或者，当 D 是由以下方式描述的 II 型平面区域时\begin{equation*}c\le y\le d,\qquad h_1(y)\le x\le h_2(y)\end{equation*}积分顺序相应地发生变化：\begin{equation*}\int_c^d\int_{h_1(y)}^{h_2(y)}\int_{u_1(x,y)}^{u_2(x,y)}f(x,y,z) dz\, dx\, dy\end{equation*}这些表示方法为在复杂的三维立体上进行积分计算提供了灵活的手段。

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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

Triple Integrals over General Regions

一般区域的三重积分

evaluating triple integrals by region

Triple integrals over bounded solid regions are used to measure a continuous function across a three-dimensional shape. The solid region E is often placed inside a rectangular box B. To handle the region cleanly, the function f(x,y,z) is extended to a new function F that matches f inside E and is zero outside the solid. The triple integral is written as ∭_E f(x,y,z) dV.

For the integral to exist, the function f must be continuous. The boundary of E also needs to be smooth enough. As with double integrals, the way the integral is evaluated depends on how the solid is described geometrically. That description is the key to choosing the order of integration.

A common case is a type I region. In this case, z lies between two continuous surfaces. The solid is written as E = {(x,y,z) | (x,y) in D; u1(x,y) ≤ z ≤ u2(x,y)}. Here, D is the projection of the solid onto the xy-plane. The lower surface is z = u1(x,y), and the upper surface is z = u2(x,y).

For a type I region, the triple integral can be rewritten as an iterated integral. The inner integral is taken with respect to z from u1(x,y) to u2(x,y), and then the result is integrated over D. If D is a type I plane region, it can be described by a ≤ x ≤ b and g1(x) ≤ y ≤ g2(x). Then the integral becomes ∫_a^b ∫_{g1(x)}^{g2(x)} ∫_{u1(x,y)}^{u2(x,y)} f(x,y,z) dz dy dx.

The same region can also be handled when the projection D is a type II plane region. In that case, D is written using c ≤ y ≤ d and h1(y) ≤ x ≤ h2(y). The order of integration changes to match that description. The iterated integral becomes ∫_c^d ∫_{h1(y)}^{h2(y)} ∫_{u1(x,y)}^{u2(x,y)} f(x,y,z) dz dx dy. These forms give flexible ways to evaluate integrals over complex three-dimensional solids.

Evaluating Triple Integrals by Region

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Evaluating Triple Integrals by Region | 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

Evaluating Triple Integrals by Region

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