Numerically Approximating Area

Suppose someone cuts a strange shape out of paper and challenges us to find the area of that paper. If we knew the shape of the boundary in terms of some equation, we might be able to set up and solve a double integral. In most cases, however, we do not have a formula for the boundary of regions, so we are forced to apporoximate the area.

The basic idea is to think of a region drawn on a sheet of graph paper. If we count the number of squares that are completely enclose within the figure, then we can compute a lower bound on the are of the region. Similarly, if we want an upper bound on the area of a region, then we can add up the areas of all boxes for which some point in the box is also in the region.

For the following exercises, you will need to load some special functions into maple with the command:

read `/u/calcIII/MVCalc.define`;

As in Lab #11, we will use an ellipse with axes 6/5 and 1 as our basic example.

Question #1

Define the maple expression:


     M := sqrt(1-(5*x/6)^2);

then issue the special Maple command:


     gridplot2d(x=-2..2,y=-1..1, grid=[10,10],region=-M..M);

Use the displayed 10x10 grid to compute a lower bound on the area of the indicated ellipse. Explain what method you used to approximate the area.
Compute an upper bound for the area. Again, explain your method.
Sketch a picture (use 2 colors!) showing the geometric significance of the two areas.

Question #2

The special Maple command leftbox2d computes the area by adding up the area of all boxes in a grid whose lower corner is contained in a region.

Compute an approximation to the area of the previous ellipse by using the command
leftbox2d(1, x=-2..2,y=-1..1, grid=[10,10], region=-M..M);
Is this area an upper bound or a lower bound for the area of the ellipse?
Will your answer to the previous question hold for regions of any shape? Argue that it will or provide a counterexample.

Next: Approximating Volume
Previous: Introduction

Frederick J. Wicklin<fjw@geom.umn.edu>

Document Created: Fri Jan 27 1995
Last modified: Tue Jan 31 13:19:29 1995