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Subtract Half, Get GI? (Posted on 2007-03-20) Difficulty: 2 of 5
If ∫o[u]x dx= ∫ou[x] dx, u>0, and [x] denotes the greatest integer < = x, is it necessarily true that u=[u]+ 1/2?

See The Solution Submitted by K Sengupta    
Rating: 2.0000 (1 votes)

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For a particular u in the range [k,k+1), where k is an integer, [u] = k, so that the first integral is the area of the isosceles right triangle with two legs equaling k.  That area is k^2/2.

The integral on the right is that of a step function. The area involved consists of 1x1 squares. When k is 1, the number of squares is zero; when k is 2, the number of squares is 1. In general, the number (and therefore area) is Sigma{i=0 to k-1} n, which is (k^2-k)/2.

However, the step function is to be evaluated beyond k, by an additional u-k to u, and the question involves how far to the right of k will result in the total area being equal to that of the isosceles right triangle of paragraph 1. To the right of x=k, the height of the step function is k. We need to gain the difference between k^2/2 and (k^2-k)/2, which is k/2.  As the height of the step function is a constant k, the base of the rectangle needs to be 1/2, to get the added area to be k/2, confirming u=[u]+1/2.

Is there a flaw in this logic?

Edited on March 20, 2007, 3:49 pm
  Posted by Charlie on 2007-03-20 15:47:49

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