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

Consider the construction of a satellite dish designed to receive communication signals. Its reflective surface has a parabolic shape and is cut at a fixed height.Engineers need to calculate the exact curved surface area to estimate protective coating requirements and fabrication costs.The rim of the dish forms a circular boundary on the ground plane, giving the domain over which the surface area is calculated. The curved surface z is modeled as a function of x and y.To find the area of this surface, a double integral is used. This formula calculates the steepness at every point by adding the squares of the partial derivatives, del z over del x and del z over del y to one.The partial derivatives of the parabolic surface are calculated and substituted into the surface area formula.Since the domain D is a circular region of radius R, the polar coordinates are used. This transformation simplifies the limits of the double integral.The double integral gives the exact reflective surface area of the dish, enabling accurate engineering design.

surface area calculations

Surface area calculations for a graph z = f(x, y) are fundamental in engineering applications involving curved structures such as satellite dishes. A parabolic dish reflects communication signals efficiently, but engineers must determine its exact curved surface area to estimate coating materials, fabrication costs, and structural requirements. Since the rim of the dish forms a circular boundary, the surface area is calculated over a circular domain in the xy-plane.Parametric Representation of the SurfaceA surface defined by z = f(x, y) can be represented parametrically as\begin{equation*}\textbf{r}(x,y)=\langle x,\; y,\; f(x,y)\rangle\end{equation*}The tangent vectors to the surface are obtained by partial differentiation:\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*}Cross Product and Surface ElementThe area of a small parallelogram on the surface is determined by the magnitude of the cross product:\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*}Expanding the determinant gives\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*}Its magnitude is\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*}Therefore, the differential surface area element is\begin{equation*}dS=\sqrt{1+\left(\jfrac{\partial z}{\partial x}\right)^2+\left(\jfrac{\partial z}{\partial y}\right)^2}\,dA\end{equation*}Surface Area FormulaAdding all differential surface elements over the domain D produces the surface area formula\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*}For the circular domain D of radius R, polar coordinates are commonly used:\begin{equation*}x=r\cos\theta,y=r\sin\theta,dA=r\,dr\,d\theta\end{equation*}where r runs from 0 to R and &#x3B8; runs from 0 to 2&#x3C0;.This transformation simplifies the limits of integration and allows the exact reflective surface area of the satellite dish to be computed accurately.

Surface Area Calculations

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Education

JoVE Core

Calculus

Unit

Multiple Integrals and Applications

KEY TERMS AND CONCEPTS

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

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the midpoint rule for double integrals

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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

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triple integrals in spherical coordinates

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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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