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 Inversely Integrating An Inverse (Posted on 2008-10-15)
Evaluate this definite integral:

1
∫ [1/y]-1 dy
0

Note: [x] is the greatest integer ≤ x.

 See The Solution Submitted by K Sengupta Rating: 3.0000 (1 votes)

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 Numerical answer | Comment 1 of 3

For purposes of taking the floor (integer part) function,

n <= 1/y < n+1

when

1/(n+1) < y <= 1/n

The width (taking y as being laid out horizontally, as the independent variable) of a given rectangle making up part of the area will be

1/n - 1/(n+1) = 1/(n(n+1))

where the height is 1/n.

This makes the area of that portion of the integral equal to

a = 1/(n^2 * (n+1))

And the total area is the sum of these for n=1 to infinity.

Summing this via computer program

DEFDBL A-Z
FOR n = 1 TO 1000000
t = t + 1 / (n * n * (n + 1))
PRINT n, t
IF t = prevT THEN
ctDown = ctDown + 1
IF ctDown > 10 THEN EXIT FOR
END IF
prevT = t
NEXT

gives a numeric answer of .6449340668423813 as the approximation.

This was reached at n = 262,102 and remained constant, to that degree of accuracy, thereafter.

 Posted by Charlie on 2008-10-15 12:28:38

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