They give a contour map. You are estimating volume using a multiple integral. Kind of like estimating the amount of dirt in a region. Construction companies do this so they can get paid for the amount of dirt they move.
This example could represent a pile of dirt that is 30 feet high and you want to find out how many cubic yards are in it.
The lines on the map are elevations. More like z coordinates.
Since the square is 4-by-4 and it says to use m=n=2, then each subrectangle has area 2-by-2.
The area of the region is 16. The area of each subrectangle is 4. So, \(\displaystyle \Delta A=4\)
Now, find a point at the center of each subrectangle and estimate the elevation there from the contours.
It would appear there is a midpoint at (1,1), (3,3), (1,3), (3,1)
Add them all up and multiply by \(\displaystyle \Delta A\)
y). Let the square R=[0
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