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Tananiajtac2

Tananiajtac2

Answered question

2022-06-03

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?

Answer & Explanation

Ashly Kaufman

Ashly Kaufman

Beginner2022-06-04Added 6 answers

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 ) = 1.2853
Compared with the exact integral result
0 1 2 x 2 = 2 + π 4 = 1.2854
the 4-point Simpson numerical integration is very accurate.
The similar procedure can be carried out for the Trapezoidal rule.
Brayan Herring

Brayan Herring

Beginner2022-06-05Added 2 answers

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'll have done the Simpson's Rule part. The Trapezoid part is similar.

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