Mathematical Tools for Physics

11—Numerical Analysis 324

11.4 Integration
The basic definition of an integral is a limit of the sum,

ξ 1 ξ 2 ξ 3 ξ 4 ξ 5

∑

f(ξi)(xi+1−xi) (xi≤ξi≤xi+1), (14)

and this is the basis for the numerical evaluation of any integral, as in section1.6.
The simplest choices to evaluate the integral off(x)over the domainx 0 tox 0 +hwould be to take the
position ofξat one of the endpoints or maybe in the middle (here I assumehis small).

∫x 0 +h

x 0

f(x)dx≈f(x 0 )h (a)

or f(x 0 +h)h (b)

or f(x 0 +h/2)h (midpoint rule) (c)

or

[

f(x 0 ) +f(x 0 +h)

]

h/ 2 (trapezoidal rule) (d)

(15)

The last expression is the average of the first two.
I can now compare the errors in all of these approximations. Setx 0 = 0.
∫h

0

dxf(x) =

∫h

0

dx

[

f(0) +xf′(0) +

1

2

x^2 f′′(0) +

1

6

x^3 f′′′(0) +···

]

=hf(0) +

1

2

h^2 f′(0) +

1

6

h^3 f′′(0) +

1

24

h^4 f′′′(0) +···.

This immediately gives the error in formula (a):

error (a) =hf(0)−

∫h

0

dxf(x)≈−

1

2

h^2 f′(0). (16)