Let \(z = f(x,y)\) define a smooth surface, and consider the corresponding parameterization \(\vr(s,t) = \langle s, t, f(s,t) \rangle\text{.}\)
Let \(D\) be a region in the domain of \(f\text{.}\) Using Equation (13.9.2), show that the area, \(S\text{,}\) of the surface defined by the graph of \(f\) over \(D\) is
\begin{equation*}
S = \iint_D \sqrt{\left(f_x(x,y)\right)^2 + \left(f_y(x,y)\right)^2 + 1} \ dA.
\end{equation*}
Use the formula developed in (a) to calculate the area of the surface defined by \(f(x,y) = \sqrt{4-x^2}\) over the rectangle \(D = [-2,2] \times [0,3]\text{.}\)
Observe that the surface of the solid describe in (b) is half of a circular cylinder. Use the standard formula for the surface area of a cylinder to calculate the surface area in a different way, and compare your result from (b).
