If f ( x ) = ⌊ x 2 ⌋ − ⌊ x ⌋ 2 ,where [x] denotes the greatest integer ≤ x then ∫ 1 2 f ( x ) d x ?

Question

If   f  (  x  )  =            ⌊              x        2            ⌋        −                  ⌊        x        ⌋            2      ,where [x] denotes the greatest integer   ≤  x then       ∫    1    2    f  (  x  )  d  x  ?

Layton Leach · Accepted Answer

On the interval (1,2) you have [x]=1. On the interval   (  1  ,      2    ) you have   [      x    2    ]  =  1 and on the interval   (      2    ,      3    ) you have   [      x    2    ]  =  3 and on the interval   (      3    ,  4  ) you have   [      x    2    ]  =  3. So you can break on the integral into:                              ∫          1          2                f        (        x        )                d        x                            =                  ∫          1                      2                          f        (        x        )                d        x        +                  ∫                      2                                3                          f        (        x        )                d        x        +                  ∫                      3                    2                f        (        x        )                d        x                                          =                  ∫          1                      2                          1        −        1                d        x        +                  ∫                      2                                3                          2        −        1                d        x        +                  ∫                      3                    2                3        −        1                d        x

If f(x)=[x^2]-[x]^2, where [x] denotes the greatest integer <=x then in_1^2 f(x)dx?

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