Approximating Volume

Up to now we've concentrated on computing areas by integrating the constant 1 over some domain. If we want the volume over a domain and beneath a surface graph, we just need to integrate the function whose graph is the surface.

In Lab #15 we modeled the metrodome by elliptical walls and a roof of the form:

roof := 1/3 - 1/6 * ((5*x/6)^2+y^2)^2;

Numerically estimate the volume of this model of the metrodome by

leftbox2d(roof,x=-2..2,y=-1..1, grid=[10,10], region=-M..M);

The volume is being estimated by rectangular blocks. The height of a block is the height of the roof over the lower left corner of the corresponding rectangle on the grid.

Question #3

Draw a sketch illustrating the geometry of this approximation to the metrodome volume. (You may want to use a very coarse grid.)
Based on your sketch, can you determine if the approximate volume is an upper or lower bound for the true volume of the model, or is it hard to tell?

Question #4

Construct a region and a surface over that region so that leftbox2d would always overestimate the volume under the surface.

Next: Convergence of Riemann Sums
Up: Introduction
Previous: Numerically Approximating Area

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

Document Created: Fri Jan 27 1995
Last modified: Tue Feb 6 10:57:46 1996