I respectfully disagree with this analysis. You cannot use an integral to reduce a sum of discrete numbers. Additionally, the number of tracks is N, therefore the sum has to run either from 1 to N or from 0 to N-1, and not from 0 to N as in the derivation (this would work for N+1 track). As a counter example, consider a disk with 3 tracks. The average seek time if you are on track 0 is 1 (assuming a uniform distribution for computing the average). If you are on track 1, the average is 2/3. Track 3 is likewise 1. Summing this up for all tracks and dividing by 3 will yield 8/9 which is less than 1/3 of the number of tracks. If you try 4 tracks, you get an average of 1.25 (4/3 = 1.33). The inaccuracy is due to the use of integration which computes the area under a curve instead of computing the sum of discrete values, and for integrating over N+1 tracks instead of N.