Simpson's rule is a method for approximating the area under a curve using the curve's ordinate and abscissa values. It divides the area into an even number of subintervals and approximates the area as the sum of weighted ordinate values. Simpson's rule provides a closer approximation to the actual area than the trapezoidal rule. The document provides the equation for Simpson's rule and an example of its application to approximate the area under a curve divided into six subintervals.

1. 25. Area under a Curve: Trapezoidal and Simpson's Rules
In(X) = -.153746 * time + 4.35568
R = .996
t (95%) = 2.776 4 d. f.
Variable Value Std. Err. Lower C. L. Upper C. L,
K: -.153746 6. 78983E-03
V-Intercept: 4.35568 .0567401
X-Intercept: 28.3303 .972837
half-life: 4.50838
X = 77.9194 * E' -.153746 t
Report: (P>rint, <S)ave, or <V>iew? (ENTER>
Computer Report
-.172595
4.19816
25.6297
4.01604
-,134898
4.51319
31. 0309
5.13832
<24> E>:ponential Growth & Decay
Pharmacologic Calculation System - Version 4.0 - 03/11/86
File name: LINE 1
Variable: time
------------- ------------- -------------
II 1 : 1 65
II 2: 2 60
II 3: 5 35
II 4: 8 25
II 5: 10 15
II 6: 15 8
In(X) -.153746 * time + 4.35568
R .996
t (95%) 2.776 4 d. f.
Variable Value Std. Err. Lower C.L. Upper C.L.
------------- ------------- ------------- ------------- -------------
K: -.153746 6. 78983E-03 -.172595 -.134898
V-Intercept: 4.35568 .0567401 4.19816 4.51319
X-Intercept: 28.3303 .972837 25.6297 31.0309
half-l i fe: 4.50838 4.01604 5.13832
X = 77.9194 * E' -.153746 t
Procedure 25
Area under a Curve: Trapezoidal and Simpson's Rules
77
Simpson's rule is a method for evaluating the area under a curve from values of
the ordinate and the abscissa. Thus, this method accomplishes the same ob-
jective as that ofthe trapezoidal rule (discussed subsequently). It may be shown,
R. J. Tallarida et al., Manual of Pharmacologic Calculations
© Springer-Verlag New York Inc. 1987

2. 78 25. Area under a Curve: Trapezoidal and Simpson's Rules
y
a b x
Figure 25.1 Area shown shaded.
however, that Simpson's rule gives a closer approximation to the area, than does
the trapezoidal rule.
The shaded region of Figure (25.1) represents the area to be measured. The
area, or integral, is S~ f(x)dx.
The interval [a, b] of the x-axis is divided into an even number (2n) of sub-
intervals, each ofwidth ~x = (b - a)/2n. The corresponding points ofthe x-axis
are Xo = a, Xl = a + ~x, X2 = a + 2 ~x, etc., and the corresponding ordinates
are Yo, Yt> etc.
The area A = J~ f(x)dx is (approximately)
A :::::! ~x/3(yo + 4Yl + 2Y2 + 4Y3 + 2Y4 + ... + 2Y2n-2 + 4Yzn-l + Yzn)·
(25.1)
Equation (25.1) is Simpson's rule.
y
3
2
o 2 3 4 5 6 x
Figure 25.2 Numerical approximation of area using six subintervals.