Manhattan College MATH 103: Simpsons Rule

Simpson's Rule

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An idea of the Simpson's rule is in following: approximate curve by parabola and then find area of
parabola (it is easy to do because we know antiderivative of quadratic function).

b−a
Again we divide [a, b] into n subintervals of equal length Δx = n
, and also require
​ n to be
even number.


Then on each consecutive pair of intervals we
approximate the curve y = f (x) by a parabola.
If y i = f (xi ), then P i = (xi , y i ) is the point on
​ ​ ​ ​ ​




the curve lying above xi . ​




A typical parabola passes through three
consecutive points P i , P i+1 and P i+2 . ​ ​ ​




First we find equation of parabola that passes
through points (x0 , y 0 ), (x1 , y 1 ) and (x2 , y 2 ). ​ ​ ​ ​ ​ ​




Also note that x1 ​ = x0 + Δx and x2 = x0 + 2Δx.
​ ​ ​




Equation of any parabola has form y = Ax2 + Bx + C and so area under parabola from x = x0 to ​




x = x2 = x0 + 2Δx is
​ ​




x +2Δx
S = ∫x00 (Ax2 + Bx + C )dx = ( A3 x3 + B 2
x + Cx)∣xx00 +2Δx =
​


​




2
​ ​ ​ ​



​
​




3 2
= ( A3 (x0 + 2Δx) +
​ ​
B
2
​ (x0 + 2Δx) + C (x0 + 2Δx)) − ( A3 x30 +
​ ​ ​ ​
B 2
x
2 0
​ ​ + Cx0 ) =
​




2 3 2
= 2AΔxx20 + 4A(Δx) x0 + 83 A(Δx) + 2BΔxx0 + 2B (Δx) + 2CΔx =
​ ​ ​ ​




Δx 2
= 3
​ (A(6x20 + 12Δxx0 + 8A(Δx) ) + B (6x0 + 6Δx) + 6C ).
​ ​ ​




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