Math 2443: Calculus & Analytic Geometry IV

Supplemental Graphics:

Surface Area

by TJ Murphy

Shows the parallelograms that are used to approximate the area of a surface; also discusses the derivation of the surface area integral from these parallelograms. (corresponding notebook for users of Mathematica who want the commands used to generate these graphics)

[Graphics:surfaceAreagr21.gif]

We start with the function [Graphics:surfaceAreagr1.gif].

[Graphics:surfaceAreagr5.gif]

To calculate the area of the surface for some restricted domain, in this case the rectangle [-1,3]x[-3,3], we first divide the rectangular domain into subrectangles.

[Graphics:surfaceAreagr7.gif]

Then we select a point in each subrectangle, in this case we chose the corner closest to the origin. For each of these points, we find the tangent plane to f at that point, and then restrict the domain of the tangent plane to the rectangle that the point came from. Two examples of these restricted tangent planes are shown in the graph below.

[Graphics:surfaceAreagr9.gif] [Graphics:surfaceAreagr10.gif]

Note that each "restricted tangent plane" has the shape of a parallelogram.

[Graphics:surfaceAreagr22.gif]

We can use the areas of the parallelograms to approximate the area of the surface. Using more and more parallelgrams gets a better and better approximation, ultimately leading to a double integral. Pick one of the parallelograms to concentrate on. Remember that it is tangent to the surface f(x,y) at a point we selected earlier. Start at this point and draw the two vectors that lie on the adjacent sides of the parallelogram. (We don't show this here because Mathematica does not draw nice arrow heads in 3D.) The cross product of these two vectors gives the area of the parallelogram. That cross product reduces nicely and leaves us with the iterated double integral for calculating the area of a surface z = f(x,y): [Graphics:surfaceArea1.gif] (assuming that the partial derivatives are continuous).

This work is part of the Multivariable Calculus with Mathematica Project at the University of Oklahoma.