Name: double integrals as cross-sections
Uploaded: 2026-06-08T09:38:57.454Z
Duration: 1 min 28 s
Channel: JoVE

二重积分用于求解函数在某一区域上的总累积量。它通过累次积分进行计算，即将二重积分表示为两个单积分的形式。对于定义在 xy 平面上矩形区域的函数，保持 x 不变意味着沿垂直于 x 轴的方向切片区域，并对 y 进行积分。此时所得结果是关于 x 的函数，再对该函数关于 x 积分即可得到总的累积量。此过程亦可逆向进行：即先沿 x 方向积分，再沿 y 方向积分。富比尼定理规范了这一过程。该定理指出，对于定义在矩形区域上的连续函数，两种积分次序将给出相同的结果。这一概念可应用于实际问题，例如求解密度变化的薄片的质量。在这种情况下，对 y 积分密度可得到一条细长条带的质量。再对这些条带的质量关于 x 进行积分，即可得到总质量。类似地，先对 x 积分再对 y 积分也会得到相同的值。富比尼定理保证了两种积分次序的结果相同，因此可以选择使计算更简便的积分次序。

累次积分与富比尼定理

二重积分将单变量函数的积分概念推广到双变量函数，从而能够计算函数 z = f(x, y) 在平面区域 R 上所围成的曲面下方的体积。对于由 a ≤ x ≤ b 和 c ≤ y ≤ d 定义的矩形区域，以及在该区域上连续的函数，二重积分可作为累次积分进行计算。这种方法通过将问题简化为依次对单个变量进行积分，从而降低了计算复杂度。迭代积分与截面面积在积分过程中保持一个变量恒定，可得到一个横截面积函数。例如，固定 x，并对函数 f(x, y) 在区间 [c, d] 上对 y 进行积分，即可定义该函数\begin{equation*}A(x) = \int_{c}^{d} f(x,y)\, dy\end{equation*}表示在位置 x 处穿过该立体的垂直切片的面积。通过沿 x 轴对该面积进行积分，即可得到曲面下的总体积：\begin{equation*}\iint_{R} f(x,y)\, dA = \int_{a}^{b} A(x)\, dx = \int_{a}^{b} \int_{c}^{d} f(x,y)\, dy\, dx\end{equation*}当首先对 x 积分，然后对 y 积分，并使用水平截面时，类似的论证同样适用。富比尼定理富比尼定理对累次积分的有效性进行了形式化。如果函数 f 在矩形区域 R 上连续，则在 R 上的二重积分可计算为：\begin{equation*}\iint_{R} f(x,y)\, dA = \int_{a}^{b} \int_{c}^{d} f(x,y)\, dy\, dx = \int_{c}^{d} \int_{a}^{b} f(x,y)\, dx\, dy\end{equation*}这一结果表明，积分顺序不会影响积分值，因此在计算平面区域内密度变化时的体积及其他物理量（如质量）时具有灵活性。

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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

Iterated Integrals and Fubini's Theorem

迭代积分与富比尼定理

double integrals as cross-sections

Double integrals help find the volume under a surface z = f(x,y) over a region in the plane. They extend the idea of a single-variable integral to functions of two variables. For a rectangular region with a ≤ x ≤ b and c ≤ y ≤ d, and for functions that are continuous on that region, the double integral can be written as an iterated integral. This breaks one harder problem into two simpler steps.

A useful way to understand this method is through cross-sectional area. If x is held constant, then integrating f(x,y) with respect to y from c to d gives a slice area function A(x) = ∫c^d f(x,y) dy. This area represents a vertical slice through the solid at position x. The total volume is then found by integrating these slice areas along the x-axis: ∬R f(x,y) dA = ∫a^b A(x) dx = ∫a^b ∫c^d f(x,y) dy dx.

The same idea also works if the order is reversed. You can hold y constant and use horizontal cross-sections, then integrate with respect to x first and y second. This gives another iterated form for the same double integral. The setup depends on whether vertical or horizontal slices are easier to compute.

Fubini’s Theorem explains why this works for continuous functions on a rectangular region. It states that the double integral over R can be computed as either ∫a^b ∫c^d f(x,y) dy dx or ∫c^d ∫a^b f(x,y) dx dy. The order of integration does not change the value. That flexibility is useful when finding volume and other quantities, such as mass when density changes across a planar region.

Double Integrals as Cross-Sections

introduction to double integrals

Double Integrals for Volume and Density

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the midpoint rule for double integrals

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iterated integrals and fubini's theorem

Double integrals help find the volume under a surface z = f(x,y) over a region in the plane. They extend the idea of a single-variable integral to functions of two variables. For a rectangular region with a ≤ x ≤ b and c ≤ y ≤ d, and for functions that are continuous on that region, the double integral can be written as an iterated integral. This breaks one harder problem into two simpler steps. A useful way to understand this method is through cross-sectional area. If x is held constant, then...

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Double Integrals on Curved Boundaries

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triple integrals in cylindrical coordinates

triple-integrals-in-cylindrical-coordinates

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triple integrals in spherical coordinates

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Finding Sphere Volume with Spherical Wedges

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change of variables in multiple integrals

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key terms and concepts

Learn how double integrals use cross-sectional area and iterated integrals to find volume under a surface, with Fubini’s Theorem for switching order.

Double Integrals as Cross-Sections | 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 as Cross-Sections

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