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].

∫ ∫ [x+y]*{x+y}*(x+y) dx dy for x = 1 to 2 and, y = 1 to 2
∫ ∫ (x+y)²[x+y] - (x+y)[x+y]² 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)²[x+y] - (x+y)[x+y]² dx for x=1 to 2
= ∫ (x+y)²*2 - (x+y)*2² dx + ∫ (x+y)²*3 - (x+y)*3² dx
= ∫ 2x²+4xy+2y²-4x-4y dx + ∫ 3x²+6xy+3y²-9x-9y dx
= (2x³/3+2x²y+2xy²-2x²-4xy| + (x³+3x²y+3xy²-9x²/2-9xy|
evaluating these using the above limits (and a lot of simplifying) gives
= 2y³/3 + 3y²/2 - 4y + 29/6
replacing the outer integral
∫ 2y³/3 + 3y²/2 - 4y + 29/6² 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