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## Numerical 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 an arbitrary surface graph, we just need to integrate the function whose graph is the surface.

Earlier in this lab, we modeled the Metrodome by elliptical walls and a roof that was a quartic polynomial. 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 8:
• 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?

QUESTION 9: Construct and sketch a region and a function on that region so that leftbox2d always overestimates the volume under the graph of the function.

Bob Hesse
Thu Feb 20 09:21:36 CST 1997