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 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.