Prove that ∫ 0 x ∫ 0 y ∫ 0 z f ( t ) d t d z d y = 1 2 ∫ 0 x ( x − t ) 2 f ( t ) d t

Question

Prove that       ∫    0    x        ∫    0    y        ∫    0    z    f  (  t  )  d  t  d  z  d  y  =      1    2        ∫    0    x    (  x  −  t      )    2    f  (  t  )  d  t

yen1291kp6 · Accepted Answer

First, take a look at the innermost double integral       ∫    0    y        ∫    0    z    f  (  t  )  d  t  d  z. This is over a triangle with t from 0 to z and z from 0 to y.If we go in the other direction, t goes from 0 to y and z goes from t to y (  t  ≤  z is the same as   z  ≥  t).So       ∫    0    y        ∫    0    z    f  (  t  )  d  t  d  z  =      ∫    0    y        ∫    t    y    f  (  t  )  d  z  d  t  =      ∫    0    y    f  (  t  )      ∫    t    y    d  z  d  t  =      ∫    0    y    f  (  t  )  (  y  −  t  )  d  tNow apply this again, and the   (  y  −  t  ) will get integrated to             (      x      −      t              )        2              2  If you have n nested integrals, this will become       ∫    0    x              f      (      t      )      (      x      −      t              )                  n          −          1                            (      n      −      1      )      !        d  x

Prove that int_0^x int_0^y f(t) dtdzdy=1/2 int_0^x (x-t)^2 f(t) dt

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