Activity 13.10.4

Let $z = f(x,y)$ define a smooth surface, and consider the parameterization $\vr(s,t) = \langle s, t, f(s,t) \rangle$.

1. Let $D$ be a region in the domain of $f$. Using equation, show that the area of the surface defined by the graph of $f$ over $D$ is $$\begin{gathered} \iint_D \sqrt{\left(f_x(x,y)\right)^2 + \left(f_y(x,y)\right)^2 + 1} \ dA \end{gathered}$$.

2. Use the formula developed in the previous part to calculate the area of the surface defined by $f(x,y) = \sqrt{4-x^2}$ over the rectangle $D = [-2,2] \times [0,3]$.

3. Observe that the surface of the solid describe the previous part 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 above.