I wanted to derive the formula for the error in the basic midpoint rule. E ( f ) = ∫ a b f [ a + b 2 , x ] ( x − a + b 2 ) d x .but I don't see how: E ( f ) = ∫ a b f [ a + b 2 , x ] ( x − a + b 2 ) d x = ( ∗ ) ?

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I wanted to derive the formula for the error in the basic midpoint rule.  E  (  f  )  =      ∫          a              b        f  [                    a        +        b            2        ,  x  ]  (  x  −                    a        +        b            2        )    d  x  .but I don&#039;t see how:  E  (  f  )  =      ∫          a              b        f  [                    a        +        b            2        ,  x  ]  (  x  −                    a        +        b            2        )  d  x  =  (  ∗  ) ?

Blaine Andrews · Accepted Answer

Step 1This is just partial integration      ∫    a    b    u      v    ′      d  x  =  u  v                    |              a    b    −      ∫    a    b        u    ′    v    d  x with   u  =  f  [            a      +      b        2    ,  x  ] and   v  =      1    2    (  x  −  a  )  (  x  −  b  )       v    ′    =  x  −            a      +      b        2  Step 2Then with      d          d      x        f  [  m  ,  x  ]  =      lim          h      →      0        f  [  m  ,  x  ,  x  +  h  ]  =  f  [  m  ,  x  ,  x  ]  =      1    2        f    ″    (  ξ  )  ,  m  =            a      +      b        2   it follows that                    E        (        f        )                            =                  ∫          a          b                f        (        x        )                d        x        −        f        (        m        )        (        b        −        a        )        =                  ∫          a          b                (        f        (        x        )        −        f        (        m        )        )                d        x                                          =                  ∫          a          b                f        [        m        ,        x        ]        (        x        −        m        )                d        x                                          =                  1          2                                      [            f            [            m            ,            x            ]            ·            (            x            −            a            )            (            x            −            b            )            ]                    a          b                −                  1          2                          ∫          a          b                f        [        m        ,        x        ,        x        ]        (        x        −        a        )        (        x        −        b        )                d        x                                          =        0        +                  1          2                f        [        m        ,        c        ,        c        ]                  ∫          a          b                (        x        −        a        )        (        b        −        x        )        d        x                                          =                                            f              ″                        (            ξ            )                    4                ·                  [          (          b          −          a          )                      ∫            a            b                    (          x          −          a          )          d          x          −                      ∫            a            b                    (          x          −          a                      )            2                              d          x          ]                                                  =                              (            b            −            a                          )              3                                24                          f          ″                (        ξ        )            with   c  ,  ξ some midpoints in the interval   [  a  ,  b  ] (more specifically,   ξ  ∈  [  m  ,  c  ] ).

I wanted to derive the formula for the error in the basic midpoint rule. E ( f ) =

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