14.8
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 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 Surface
A 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 Element
The 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} & \textbf{j} & \textbf{k}\\1 & 0 & \jfrac{\partial z}{\partial x}\\0 & 1 & \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 Formula
Adding 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 θ runs from 0 to 2π.
This transformation simplifies the limits of integration and allows the exact reflective surface area of the satellite dish to be computed accurately.
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.
From Chapter 14:
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