Name: calculating curved dish surface area
Uploaded: 2026-06-08T09:38:55.730Z
Duration: 1 min 22 s
Channel: JoVE

考虑构建一个用于接收通信信号的卫星天线。其反射面呈抛物面形状，并在固定高度处被截断。工程师需要计算精确的曲面面积，以估算防护涂层的需求量和制造成本。培养皿的边缘在基平面上形成一个圆形边界，该边界定义了表面积的计算区域。曲面 z 被建模为 x 和 y 的函数。为了求出该曲面的面积，需使用二重积分。该公式通过将偏导数 ∂z/∂x 和 ∂z/∂y 的平方与 1 相加，计算出曲面上每一点的陡峭程度。计算抛物面的偏导数，并将其代入表面积公式中。由于区域 D 是半径为 R 的圆形区域，因此采用极坐标。该变换可简化二重积分的积分限。二重积分给出了碟形反射面的精确表面积，从而实现准确的工程设计。

表面积计算

Surface Area Calculations

对于形如 z = f(x, y) 的曲面，其表面积计算在涉及卫星天线等曲面结构的工程应用中具有基础性作用。抛物面天线能够高效反射通信信号，但工程师必须确定其精确的曲面面积，以估算涂层材料、制造成本和结构需求。由于天线边缘在 xy 平面上形成圆形边界，因此表面积的计算需在该圆形区域内进行。曲面的参数表示由 z = f(x, y) 定义的曲面可以参数化表示为\begin{equation*}\textbf{r}(x,y)=\langle x,\; y,\; f(x,y)\rangle\end{equation*}通过偏微分得到曲面的切向量：\begin{equation*}\textbf{r}_x=\left\langle1,\;0,\;\jfrac{\partial z}{\partial x}\right\rangle\end{equation*}\begin{equation*}\textbf{r}_y=\left\langle0,\;1,\;\jfrac{\partial z}{\partial y}\right\rangle\end{equation*}叉积与面元曲面上一个小平行四边形的面积由叉积的模长决定：\begin{equation*} \textbf{r}_x\times \textbf{r}_y = \left| \begin{array}{ccc} \textbf{i} &amp; \textbf{j} &amp; \textbf{k}\\ 1 &amp; 0 &amp; \jfrac{\partial z}{\partial x}\\ 0 &amp; 1 &amp; \jfrac{\partial z}{\partial y} \end{array} \right| \end{equation*}展开行列式可得\begin{equation*}\textbf{r}_x\times \textbf{r}_y=\left\langle -\jfrac{\partial z}{\partial x}, -\jfrac{\partial z}{\partial y}, 1 \right\rangle\end{equation*}其大小为\begin{equation*}\left|\textbf{r}_x\times \textbf{r}_y\right|=\sqrt{1+\left(\jfrac{\partial z}{\partial x}\right)^2+\left(\jfrac{\partial z}{\partial y}\right)^2}\end{equation*}因此，微分表面积元素为\begin{equation*}dS=\sqrt{1+\left(\jfrac{\partial z}{\partial x}\right)^2+\left(\jfrac{\partial z}{\partial y}\right)^2}\,dA\end{equation*}表面积公式对区域 D 内所有微分表面元素求和，得到表面积公式\begin{equation*}A(S)=\iint_D\sqrt{1+\left(\jfrac{\partial z}{\partial x}\right)^2+\left(\jfrac{\partial z}{\partial y}\right)^2}\,dA\end{equation*}对于半径为 R 的圆形区域 D，通常使用极坐标：\begin{equation*}x=r\cos\theta,y = r\sin\theta,dA = r dr dθ\end{equation*}其中 r 从 0 到 R，θ 从 0 到 2π。该变换简化了积分限，从而能够准确计算卫星天线的精确反射表面积。

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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

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 parametric form as r(x,y) = . The tangent vectors on the surface come from partial differentiation. They are r_x = and r_y = . These vectors describe how the surface changes in the x- and y-directions.

The area of a small piece of the surface comes from the cross product of those tangent vectors. The determinant form is r_x × r_y = | i j k; 1 0 ∂z/∂x; 0 1 ∂z/∂y |. Expanding it gives r_x × r_y = . Its magnitude is sqrt(1 + (∂z/∂x)^2 + (∂z/∂y)^2), which gives the differential surface area element dS = sqrt(1 + (∂z/∂x)^2 + (∂z/∂y)^2) dA.

Adding all of the small surface elements over the domain D gives the surface area formula A(S) = ∬_D sqrt(1 + (∂z/∂x)^2 + (∂z/∂y)^2) dA. For a circular domain of radius R, polar coordinates are often used. The change of variables is x = r cosθ, y = r sinθ, and dA = r dr dθ. The limits are r from 0 to R and θ from 0 to 2π. This setup makes it easier to compute the exact reflective surface area of the satellite dish.

Calculating Curved Dish Surface Area

introduction to double integrals

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.

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 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...

substitutions in multiple integrals

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

Multiple integrals can be used to find the volume of curved three-dimensional objects. This method is useful for shapes such as storage tanks, pipes, and reservoirs. It works well when the region changes shape across space. Cylindrical coordinates are especially helpful for objects with rotational symmetry. In this system, each point is described by the radial distance r, the angle θ, and the vertical coordinate z. These coordinates make circular and paraboloid-shaped regions easier to...

key terms and concepts

Learn how surface area formulas, partial derivatives, and polar coordinates are used to find the curved area of a satellite dish over a circular domain.

Calculating Curved Dish Surface Area | 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

Calculating Curved Dish Surface Area

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