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 n1 to n.
∫ [a]*{a}*(a)
da =
∫ a²[a] + a[a]²
da from n1 to n
now on this interval [a] is just a constant (n1)
(n1)
∫ a²  a(n1)
da from n1 to n =
(n1) (a³/3  a²(n1)/2 from n1 to n =
(n1) (n³/3  (n1)³/3  n²(n1)/2 + (n1)²(n1)/2) =
(n1)(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 20110725 00:11:00 