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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

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