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 3y where [x+y]=2
the second from 3y 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+2y²4x4y
dx +
∫ 3x²+6xy+3y²9x9y
dx= (2x³/3+2x²y+2xy²2x²4xy + (x³+3x²y+3xy²9x²/29xy
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 20110725 19:03:12 