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].

Instead of using two variables of integration lets just drop it down to one.  Lets integrate [a]*{a}*a da from 1 to n, with n a positive integer.

Since {a} = a-[a] we can substitute this into the expression to be integrated which simplifies to a²[a] - a[a]²

Now rather than try to integrate the whole discontinuous expression lets just integrate it from n-1 to n.

∫ [a]*{a}*(a) da =
∫ a²[a] + a[a]² da from n-1 to n
now on this interval [a] is just a constant (n-1)
(n-1)∫ a² - a(n-1) da from n-1 to n =
(n-1) (a³/3 - a²(n-1)/2| from n-1 to n =
(n-1) (n³/3 - (n-1)³/3 - n²(n-1)/2 + (n-1)²(n-1)/2) =
(n-1)(n/2 - 1/6) =
(3n² - 4n + 1)/6

Now to get from 1 to n just sum:
Σ (3n² - 4n + 1)/6 from 1 to n =
(2n³ - n² - n)/12

The two variable solution sought is quite a bit more complicated since x+y can abruptly exceed the next integer as a less predicable time while integrating.  I don't see how a similar method to this would work.

Posted by Jer on 2011-07-25 00:11:00