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
y = ax2 + bx + c
(0, y1 )
(−h, y0 )
(h, y2 )
y1
y2
y0
∆x
0
−h
Z
h
x
h
(ax2 + bx + c) dx
A=
−h
=
h
bx2
ax3
+
+ cx 3
2
−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 LATEX on April 23, 2008.
We consider the definite integral
Z
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 + 2∆x,
...,
xn = a + n∆x = b.
We can compute the value of f (x) at these points.
y0 = f (x0 ),
y1 = f (x1 ),
y2 = f (x2 ),
...,
yn = f (xn ).
y
y4
y0
y1
a = x0 x1
y2
x2
yn
y3
x3
yn−2
x4
···
∆x
yn−1
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
Z
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.
Z
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
Z 4p
1 + x3 dx.
1
Solution. For n = 6, we have ∆x =
x
√
y = 1 + x3
4−1
6
1
√
2
= 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
Therefore,
Z
1
4
p
√
√
√
√
√ 0.5 √
1 + x3 dx ≈
2 + 4 4.375 + 2(3) + 4 16.625 + 2 28 + 4 43.875 + 65
3
≈ 12.871