Find the values of Riemann sums which are reasonable over- and under-estimates f

Question

Find the values of Riemann sums which are reasonable over- and under-estimates for

f(x,y) values

Let R be the rectangle 2?x?2.4, 4?y?4.6. Find the values of Riemann sums which are reasonable over- and under-estimates for ?Rf(x,y)dA with ?x=0.2 and ?y=0.3

I feel so retarted right now. i know that the over-estimate will be the sum of the area of each square times the largest value the function takes at any of the corners any sub-rectangle but...what squares are we talking about here? how do you know which min/max points to pick from the table? Can you please explain, Thanks

Explanation / Answer

The first subrectangle is from x=2 to x=2.2 and from y=4 to y=4.3. This is the 2x2 square in the top left of the table, so the corners have values of 6,7,5,and 6. The largest value of f in this subrectangle is 7.

The next is the 2x2 square in the top right so from 2.2 to 2.4 and 4 to 4.3. The largest is 9.

Then in the bottom left(2 to 2.2 and 4.3 to 4.6) the largest is 6.

And finally the bottom right has largest value of 13.

Each subrectangle has a base area of 0.2x0.3, which equal .06. So we multiply .06*(7+9+6+13)=2.1 to get the over estimate. To underestimate we use the same rectangles, but use the smallest and so we have .06*(5+6+4+5)=1.2

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