Approximate ∫ 0 1 2 − x 2 d x using the trapezoidal and simpson's rule for 4 intervals.Now we can determine the simpson rule is h 3 ( f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + f ( x 4 ) ) and the trapezoidal rule is h 2 ( f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + 2 f ( x 3 ) + f ( x 4 ) ) and h = b − a n which I assume is 1 − 0 4 but how we add it all together?

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Approximate       ∫    0    1          2    −          x      2        d  x using the trapezoidal and simpson&#039;s rule for   4 intervals.Now we can determine the simpson rule is      h    3              (        f  (      x    0    )  +  4  f  (      x    1    )  +  2  f  (      x    2    )  +  4  f  (      x    3    )  +  f  (      x    4    )            )      and the trapezoidal rule is      h    2              (        f  (      x    0    )  +  2  f  (      x    1    )  +  2  f  (      x    2    )  +  2  f  (      x    3    )  +  f  (      x    4    )            )      and   h  =            b      −      a        n   which I assume is             1      −      0        4  but how we add it all together?

Ashly Kaufman · Accepted Answer

Take the Simpson rule as an example.Pick the equally-spaced points at   0  ,  0.25  ,  0.5  ,  0.75 and   1. Let   f  (  x  )  =      2    −          x      2      . Then, the Simpson expression becomes,      1    12    (      2    +  4      2    −          0.25      2        +  2      2    −          0.5      2        +  4      2    −          0.75      2        +  1  )  =      1    12    (      2    +      31    +      7    +      23    +  1  )  =  1.2853Compared with the exact integral result      ∫    0    1        2    −          x      2        =            2      +      π        4    =  1.2854the 4-point Simpson numerical integration is very accurate.The similar procedure can be carried out for the Trapezoidal rule.

Brayan Herring · Accepted Answer

You need to actually find the values of       x    0  ,       x    1  , …,       x    4   and then plug them into   f so that you get the values of   f  (      x    0    ),   f  (      x    1    ), …,   f  (      x    4    ). Then you just plug them in and evaluate.For example,       x    0    =  0 and       x    1    =  0  +      1    4  , so   f  (      x    0    )  =      2    −          0      2        =      2   and   f  (      x    1    )  =      2    −                  (                  1          4                )            2        =            31        4  . Then plug those in. For example, with Simpson, you get      h    3        (    0    +    4    ⋅                  31            4        +    2    f    (          x      2        )    +    4    f    (          x      3        )    +    f    (          x      4        )    )  Now do it for       x    2  ,       x    3  , and       x    4  , and you&#039;ll have done the Simpson&#039;s Rule part. The Trapezoid part is similar.

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