Calculus: Analytic Geometry and Calculus, with Vectors

4.9 Simpson and other approximations to integrals 281

3 Using the Simpson formula with it = 2, obtain the approximations

and

4

12.57 1 dx 031 [ 215 +2 6 + 217 =

0.07696 106

22.718 1 0.0 9 1+ 4+

2.7 Xdx= 3 [2.7 2.709 2.718^1 0.00664 454.

Use these formulas and the first formula of Problem 2 to obtain the approximation

log 2.718 = 0.99989 633.

Remark: With a little skill and a desk calculator that makes divisions, it is not

difficult to extend these calculations to obtain good approximations to the number

e = 2.71828 18284 59045

for whichf 1e z dx= 1 and log e = 1. Better ways to approximate logarithms

and e will appear later.

4 Someday we will learn the formulas

1+1

x2dx=tan'x+c, fo11+x^dx=4=0.7853981634.

Use the Simpson formula to find approximations to the last of these integrals,

and find the errors in the approximations, to obtain the numbers in the first two

or three rows of the following table.

it Simpson value Error

2 .78333 332 .00206 484

4 .78539 212 .00000 604

6 .78539 782 .00000 034

8 .78539 802 .00000 014

10 .78539 809 .00000 007

12 .78539 812 .00000 004

14 .78539 812 00000 004

16 78539 809 .00000 007

18 .78539 812 .00000 004

20 .78539 809 .00000 007

40 .78539 789 .00000 027

60 .78539 782 .00000 034

80 .78539 782 .00000 034

100 .78539 769 .00000 047

200 .78539 769 .00000 047

400 .78539 569 .00000 247

600 .78539 465 .00000 351

800 .78539 425 .00000 391

1000 .78539 395 .00000 421

10000 .78535 725 .00004 091

15000 .78535 442 .00004 374

20000 .78535 265 .00004 551

100000 .78499 059 .00040 757