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Floors and decimals integral II (Posted on 2011-07-21) Difficulty: 3 of 5
Each of m and n is a positive integer with m < n. Evaluate this double definite integral in terms of m and n.

[x+y]*{x+y}*(x+y) dx dy for x = 1 to m and, y = 1 to n

*** [x] denotes the smallest integer ≤ x, and {x} is the decimal part of x, ie {x}=x-[x].

No Solution Yet Submitted by K Sengupta    
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Solution for m=n=2 | Comment 2 of 3 |
[x+y]*{x+y}*(x+y) dx dy for x = 1 to 2 and, y = 1 to 2
(x+y)[x+y] - (x+y)[x+y] dx dy for x = 1 to 2 and, y = 1 to 2

The inner integral needs to be done in two pieces
the first from 1 to 3-y where [x+y]=2
the second from 3-y to 2 where [x+y]=3
for clarity I will not try to write in these limits of integration and I will leave off the outer integral until the end

(x+y)[x+y] - (x+y)[x+y] dx for x=1 to 2
= (x+y)*2 - (x+y)*2 dx + (x+y)*3 - (x+y)*3 dx
= 2x+4xy+2y-4x-4y dx + 3x+6xy+3y-9x-9y dx
= (2x/3+2xy+2xy-2x-4xy| + (x+3xy+3xy-9x/2-9xy|
evaluating these using the above limits (and a lot of simplifying) gives
= 2y/3 + 3y/2 - 4y + 29/6
replacing the outer integral
2y/3 + 3y/2 - 4y + 29/6 dy for y=1 to 2
= 29/6

Barring an algebra error the solution for m=n=2 is 29/6 


  Posted by Jer on 2011-07-25 19:03:12
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