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.

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

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

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

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

Trapezoid Approximation of a Definite Integral where …

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

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.

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.

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.

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

Simpson’s Rule

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

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:

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

Determining the Number of Intervals (n) Find n so that the midpoint method will have an error less than 10 -4.