What is the arc length of f ( x ) = x 3 6 + 1 2 x on x ∈ [ 1 , 3 ]?

Question

What is the arc length of   f  (  x  )  =            x              3              6    +      1          2      x       on   x  ∈  [  1  ,  3  ]?

Turynka2f · Accepted Answer

If   f  :  x  ↦            x              3              6    +      1          2      x      , the length is  L  =      ∫          1              3            1    +          f      ′        (    x          )              2              d  xBecause       f    ′    (  x  )  =            x              2              2    −      1          2              x                  2                      =                    x                  4                    −      1              2              x                  2                    , you have      f    ′    (  x      )          2        +  1  =                    x                  8                    −      2              x                  4                    +      1              4              x                  4                      +  1  =                    x                  8                    +      2              x                  4                    +      1              4              x                  4                      =  (                    x                  4                    +      1              2              x                  2                          )          2      so, you can write  L  =      ∫          1              3                          x                  4                    +      1              2              x                  2                      d  x  =      ∫          1              3        (            x              2              2    +      1          2              x                  2                      )  d  x  L  =  [            x              3              6    −      1          2      x            ]          1              3        =      14    3

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