Estimation of double integral

[engineertobe]

How do I even go about this?

The figure below shows level curves of a function f(x

y). Let the square R=[0

4]

[0

4]. Estimate the value of

Rf(x

y)dA with the Midpoint rule and m=n=2. (Be careful to notice what R is defined to be!)

 

[galactus]

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 \(\displaystyle \Delta A\)

 

[theodorelockwood]

Yeah, estimating dirt volume with contour maps is just like what we do in takeoffs it is all about breaking it into sections and calculating accurately. Using midpoints to estimate elevations and multiplying by ΔA gives you a solid volume estimate, just like construction estimating consultants do for earthwork projects. Precision here is key because even small miscalculations can mean big cost differences!