Name: double integrals for volume and density
Uploaded: 2026-06-08T09:38:14.541Z
Duration: 1 min 28 s
Channel: JoVE

二重积分给出了由二元函数在其定义域上所定义曲面下方的体积，该区域在平面上显示。假设一个挖掘区域由 xy 平面上的一个矩形区域定义。该区域上方存在一个曲面，其高度表示地面海拔。需要挖掘的土壤总体积是该表面与矩形底面之间的空间。为了计算该体积，将底面划分为许多小的子矩形，每个子矩形的面积为 ΔA。对于每个子矩形，选择一个样本点，该点处的函数值即为高度。将此高度乘以 ΔA 可形成一个垂直的矩形土壤柱体。将网格中所有列的体积相加可得到一个近似的总体积。这种近似方法称为二重黎曼和。当固定区域内子矩形的数量增加时，该近似结果会更加精确。当网格无限细化，且每个子矩形的面积趋近于零时，取极限即定义了二重积分。二重积分表示曲面在该区域上方所围成的精确体积。

双重积分简介

Introduction to Double Integrals

Double integrals are fundamental tools in multivariable calculus for computing volumes under surfaces defined by functions of two variables. While single-variable integration calculates the area under a curve, double integrals extend this concept to evaluate the volume between a surface and a planar region. This generalization is particularly valuable in physical and engineering contexts, where many systems and structures are best modeled in three dimensions.In single-variable integration, the area under a curve y = f(x) over an interval [a, b] is computed by partitioning the interval into subintervals, evaluating the function at representative points, and summing the areas of thin rectangles. This process leads to the definite integral:\begin{equation*}\int_a^b f(x),dx = \lim_{n\to\infty}\sum_{i=1}^{n} f(x_{\textit{i}}),\Delta x\end{equation*}For functions of two variables, f(x, y), defined over a region R in the xy-plane, the analogous concept involves calculating the volume beneath the surface z = f(x, y) and above the region R. This is accomplished using a double integral.To approximate the volume, the region R is subdivided into smaller subregions, typically rectangles. At each subregion, a sample point (xij , yij) is selected, and the value f(xij , yij) gives the height of a vertical column above that subregion. The volume of each column is approximated by the product of the height and the subregion&#x2019;s area. Summing over all subregions yields a double Riemann sum. Taking the limit as the size of the subregions approaches zero defines the double integral:\begin{equation*}\iint_R f(x,y)\,dA = \lim_{m,n\to\infty}\sum_{i=1}^{m}\sum_{j=1}^{n} f(x_{\textit{ij}},y_{\textit{ij}})\,\Delta A\end{equation*}which represents the exact volume under the surface and over the region R.Double integrals are widely used in applications involving physical quantities such as mass, charge, and fluid volume distributed over a region. When the integrand f(x, y) represents a density function, the double integral yields the total quantity over the region. Thus, double integrals are powerful tools for analyzing and modeling multivariable systems in mathematics, physics, and engineering.

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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

双积分导论

double integrals for volume and density

Double integrals help find the volume under a surface and over a region in the xy-plane. They extend single-variable integration, which finds area under a curve on an interval, to functions of two variables. This makes them useful for models in mathematics, physics, and engineering that need three-dimensional thinking.

Single-variable integration uses thin rectangles to approximate area under y = f(x) from a to b. The definite integral is written as a limit of a sum of these rectangle areas. Double integrals use the same basic idea, but they work on a region R in the xy-plane instead of a line segment.

For a function f(x, y), the goal is to measure the volume beneath the surface z = f(x, y) and above region R. The region is divided into smaller subregions, often rectangles. At each subregion, a sample point (xij, yij) is chosen, and f(xij, yij) gives the height of a vertical column above that piece.

The volume of each column is estimated by multiplying the height by the area of the subregion. Adding all of those pieces gives a double Riemann sum. Taking the limit as the subregions become very small defines the double integral, written as ∬R f(x,y) dA, which gives the exact volume over R.

Double integrals also apply to physical quantities spread across a region. If the integrand f(x, y) is a density function, the double integral gives the total amount, such as mass, charge, or fluid volume. This makes double integrals a powerful tool for analyzing multivariable systems.

Double Integrals for Volume and Density

introduction to double integrals

Double integrals help find the volume under a surface and over a region in the xy-plane. They extend single-variable integration, which finds area under a curve on an interval, to functions of two variables. This makes them useful for models in mathematics, physics, and engineering that need three-dimensional thinking. Single-variable integration uses thin rectangles to approximate area under y = f(x) from a to b. The definite integral is written as a limit of a sum of these rectangle areas.

the midpoint rule for double integrals

Estimating Surface Volume with Midpoints

The midpoint rule estimates volume for a double integral over a rectangular region. It is useful when a surface has a continuously changing height. In civil engineering, it can help estimate how much soil must be moved when planning a road across uneven terrain. The road footprint is treated as a rectangle in the xy-plane. The terrain above a flat reference level is described by a continuous height function, f(x, y). The goal is to estimate the volume of soil between that surface and the...

iterated integrals and fubini's theorem

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...

average value on a rectangle

Estimating Snowfall with Midpoint Rule

Estimating snowfall with the midpoint rule uses a rectangular region and a snow depth function of two variables. A snowfall map can show how snow depth changes across an area, such as Colorado, over a fixed time period. If f(x,y) gives the snow depth at a point in the region, then the average snowfall is found by comparing the total snowfall with the area of the region. The total snowfall over the rectangle can be written as a double integral. This integral adds the snowfall from every point...

double integrals over general regions

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...

changing the order of integration in double integrals

changing-the-order-of-integration-in-double-integrals

Finding Volume with Double Integral Bounds

Double integrals can be used to find the volume of liquid in tanks with irregular sides. The base of the tank is divided into many very small rectangular sections. Each section becomes the base of a thin column of liquid, and the volumes of all the columns are added across the region. This gives an accurate estimate of the total liquid volume inside the tank. The limits of integration come from the boundaries of the tank’s base. These limits depend on how the region is divided during the...

double integrals in polar coordinates

Polar Area Elements for Double Integrals

Double integrals use polar coordinates to measure area and other quantities over two-dimensional regions with circular symmetry. This approach is useful for curved shapes that appear in engineering and design, such as ponds, reservoirs, and circular foundations. Polar coordinates describe each point by its distance from the center and its angle, which often makes the region easier to set up for integration. A semicircular pond with radius L shows why polar coordinates can be simpler than...

surface area calculations

Calculating Curved Dish Surface Area

Calculating curved dish surface area is an important step in engineering for shapes like satellite dishes. A parabolic dish can reflect communication signals efficiently, but engineers still need its exact curved surface area. That value helps them estimate coating material, fabrication costs, and structural requirements. Because the rim of the dish makes a circular boundary, the calculation is set up over a circular domain in the xy-plane. A surface given by z = f(x, y) can be written in...

triple integrals in rectangular coordinates

triple-integrals-in-rectangular-coordinates

Triple Integrals for Volume and Mass

Triple integrals help find the total value of a function across a three-dimensional region. They are often used to calculate volume, mass, and other physical quantities that change from place to place. The idea is to split a solid region into many small rectangular boxes, evaluate the function in each box, and add the results. As the boxes get smaller, the triple Riemann sum gets closer to the exact triple integral. In rectangular coordinates, the integral is written as an iterated integral,...

triple integrals over general regions

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...

changing the order of integration in triple integrals

changing-the-order-of-integration-in-triple-integrals

Setting Triple Integral Bounds for Easier Evaluation

Triple integrals can be easier to evaluate when the bounds are chosen to match the shape of the solid. In this example, the region is enclosed by a flat base, a slanted plane, two vertical planes, and a parabolic cylinder. The task is to integrate e x over this three-dimensional region, so the goal is to describe the limits in the simplest useful order. One setup uses x as the innermost variable. In that case, each line segment inside the solid runs in the x-direction. The lower limit for x...

triple integrals in cylindrical coordinates

triple-integrals-in-cylindrical-coordinates

Volume of Cylinders with Triple Integrals

Cylindrical coordinates describe a point in three-dimensional space with three values: radial distance, angle, and height. The height gives the position above the xy-plane. The radial distance tells how far the point is from the z-axis, and the angle gives its direction from the positive x-axis in the xy-plane. This system is useful for regions with circular symmetry. It matches the shape of cylinders, disks, and circular tanks. For that reason, it is a natural choice when finding volume in...

triple integrals in spherical coordinates

triple-integrals-in-spherical-coordinates

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

change-of-variables-in-multiple-integrals

Using the Jacobian to Simplify Integrals

Multiple integrals are used to find areas, volumes, mass distributions, and other physical quantities over regions in two or three dimensions. In many problems, the original region has curved boundaries when written in Cartesian coordinates. Those boundaries can make the limits of integration hard to describe and the calculation more difficult. A change of variables helps simplify the setup. It replaces the original coordinates x and y with new variables u and v. The original region R in the...

moments and centers of mass

Balance Point of a Variable-Density Plate

A flat metal plate, or lamina, balances at a point called its center of mass. That balance point depends on the plate’s shape and on how mass is spread through the material. If the density changes from place to place, some regions affect the balance more than others. The center of mass can be treated as a weighted average of every point in the plate. For a plate in the xy-plane, the density at each point is written as ρ(x, y). To study the mass distribution, the region D is divided into many...

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Changing Variables in Double Integrals

Changing variables in double integrals can make a hard region much easier to integrate over. This method is useful when finding physical quantities spread across a two-dimensional area, such as the total mass of an elliptical plate. The density function is evaluated at every point in the region, and all of those contributions are added together to get the mass. If the integral is set up in rectangular coordinates, an ellipse often creates difficult limits. Those limits can involve square root...

real-life applications of multiple integrals

real-life-applications-of-multiple-integrals

Volume of Curved Tanks with Multiple Integrals

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key terms and concepts

Learn how double integrals estimate volume under a surface, build from double Riemann sums, and measure totals like mass, charge, or fluid volume.

Double Integrals for Volume and Density | 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 for Volume and Density

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