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Simpson's Rule
Simpson's rule is a numerical method that approximates the value of a definite integral by using quadratic polynomials. Let's first derive a formula for the area under a parabola of equation y = ax2 + bx + c passing through the three points: (-h, y0 ), (0, y1 ), (h, y2 ). y (0, y1 ) y = ax2 + bx + c (h, y2 )
(-h, y0 ) y0
y1
y2 x
-h
0
h
x
h
A=
-h
(ax2 + bx + c) dx bx2 ax3 + + cx 3 2
h
=
-h
2ah3 = + 2ch 3 h 2ah2 + 6c = 3 Since the points (-h, y0 ), (0, y1 ), (h, y2 ) are on the parabola, they satisfy y = ax2 + bx + c. Therefore, y0 = ah2 - bh + c y1 = c y2 = ah2 + bh + c Observe that y0 + 4y1 + y2 = (ah2 - bh + c) + 4c + (ah2 + bh + c) = 2ah2 + 6c. Therefore, the area under the parabola is A= h x (y0 + 4y1 + y2 ) = (y0 + 4y1 + y2 ) . 3 3
Gilles Cazelais. Typeset with L T X on April 23, 2008. A E

We consider the definite integral
b
f (x) dx.
a
We assume that f (x) is continuous on [a, b] and we divide [a, b] into an even number n of subintervals of equal length b-a x = n using the n + 1 points x0 = a, x1 = a + x, x2 = a + 2x, ..., xn = a + nx = b.
We can compute the value of f (x) at these points. y0 = f (x0 ), y y1 = f (x1 ), y2 = f (x2 ), ..., yn = f (xn ).
y0 y1 y2
y4 y3 yn-2 yn-1
yn
a = x0 x1
x2
x3
x4
···
x
···
xn-2 xn-1 xn = b
x
We can estimate the integral by adding the areas under the parabolic arcs through three successive points.
b
f (x) dx 
a
x x x (y0 + 4y1 + y2 ) + (y2 + 4y3 + y4 ) + · · · + (yn-2 + 4yn-1 + yn ) 3 3 3
By simplifying, we obtain Simpson's rule formula.
b
f (x) dx 
a
x (y0 + 4y1 + 2y2 + 4y3 + 2y4 + · · · + 4yn-1 + yn ) 3
Example. Use Simpson's rule with n = 6 to estimate
4
1 + x3 dx.
1
Solution. For n = 6, we have x = x  y = 1 + x3 Therefore,
4
4-1 6
= 0.5. We compute the values of y0 , y1 , y2 , . . . , y6 .  1.5 4.375 2 3  2.5 16.625 3  28  3.5 43.875 4  65
1  2
1 + x3 dx 
1
     0.5  2 + 4 4.375 + 2(3) + 4 16.625 + 2 28 + 4 43.875 + 65 3
 12.871