Determine the value of the
positive integer constant A that satisfies this relationship:
6
∫ {y}
^{[y]} dy = 19/20
A
Bonus Question:
What are the possible pair(s) of the
positive integer constants (B, C) that satisfy this relationship?
C
∫ [y]
^{{y}} * ln [y] dy = 18
B
Notes:
(i) [x] is the greatest integer ≤ x, and {x}= x  [x]
(ii) ln x is the
natural logarithm of x.
Because of their breaks at integer values, the functions are best integrated piecewise, between integral values of y.
For a given unit interval of y, where [y] = k, the first expression has the value
Integ{0 to 1} x^k dx
where x={y} but in my Integ expression, the {} do not have the given meaning, but merely show the limits of integration.
The evaluation of the above integral is
(x^(k+1))/(k+1)  0 to 1
= 1/(k+1)
when k (that is, [y]) takes on the following values, the integral for the unit interval is:
k integral
1 1/2
2 1/3
3 1/4
4 1/5
5 1/6
If A were 1, the integral would be the total of these: 29/20. We need to get rid of 10/20 or 1/2, and can do this be removing the k=1 interval. So the integration starts at k=2, or y=2, and thus A = 2.
Bonus:
In the same way, using unit intervals of y, where [y]=k, we add up, from B to C1 (as we're dealing with the bottom integer of each unit interval), the evaluations of
Integ{0 to 1} k^x * (ln k) dx
This integral is k^x  0 to 1
= k  1
So we need a B and C such that
Sigma{k = B to C1} (k  1) = 18
or Sigma{i = B1 to C2} i = 18
The lower and upper limits for the Sigma consistent with evaluating to 18, could be, with the corresponding B and C values:
i values B C
18 to 18 19 20
5 to 7 6 9
3 to 6 4 8
so (B,C) is (19,20), (6,9) or (4,8)

Posted by Charlie
on 20080516 16:45:37 