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 #11 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 of the true volume of the model?

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: Mon Jan 30 15:50:13 1995