1. MTH 252 Integral Calculus Chapter 8 – Principles of
Integral Evaluation
Section 8.7 – Numerical Integration;
Simpson’s Rule
Copyright © 2006 by Ron Wallace, all rights reserved.

2. Reminder: Definition of a Definite Integral
where …
Since any sequence of partitions may be used provided max xk 0, using regular partitions …

3. Equivalent Definition of a Definite Integral
where …
Natural choices for xk* include:
left endpoint = xk-1
right endpoint = xk
midpoint = (xk-1+xk)/2

4. Midpoint Approximation of a Definite Integral
where … a b m1 m2 m3 m4

5. Midpoint Approximation of a Definite Integral
Example!
Approximate w/ n = 4 11

6. Trapezoid Approximation of a Definite Integral
where …

7. Trapezoid Approximation of a Definite Integral
Example!
Approximate w/ n = 4 10
4+2(10)-4 = 20

8. Simpson’s Rule f(x) x x+h x-h y1 = f(x-h) y2 = f(x) y3 = f(x+h)
p(x) = ax2+bx+c
y1 = f(x-h)
y2 = f(x)
y3 = f(x+h)
Three non-linear points determine a unique parabola.

9. Simpson’s Rule h -h x+h x-h x y1 = f(x-h) y2 = f(x) y3 = f(x+h)
p(x) = ax2+bx+c h -h
x+h
x-h x
y1 = f(x-h)
y2 = f(x)
y3 = f(x+h)
Three non-linear points determine a unique parabola.

10. Simpson’s Rule x h -h y1 = f(x-h) y2 = f(x) y3 = f(x+h) (0, y2)
p(x) = ax2+bx+c
(0, y2) x h -h
(h, y3)
(-h, y1)
y1 = f(x-h)
y2 = f(x)
y3 = f(x+h)
Three non-linear points determine a unique parabola.

11. Simpson’s Rule f(x) x1 x2 x3 (x2 , y2) (x1 , y1) (x3 , y3)
p(x) = ax2+bx+c x1 x2 x3

12. Simpson’s Rule

13. Simpson’s Rule Approximation
Example!
Approximate w/ n = 4 6
4+4(6)+2(4)-4 = 32

14. Error Estimation Midpoint Method: Trapezoid Method: Simpson’s Rule:
Let
What happens when you double the number of intervals?
Midpoint Method:
Trapezoid Method:
Simpson’s Rule:

15. Error Estimation Example! Midpoint Method  11 Trapezoid Method  10
Approximated w/ n = 4
Midpoint Method  11 3 2 ) 4 ( 12 = - n K a b E T
Trapezoid Method  10
Simpson’s Rule  10-2/3

16. Determining the Number of Intervals (n)
Find n so that the midpoint method will have an error less than