Computational Physics - Department of Physics

110 5 Numerical Integration

✲

f(x)

x

✻

a a+h a+ 2 h a+ 3 h b

Fig. 5.1The area enscribed by the functionf(x)starting fromx=atox=b. It is subdivided in several smaller

areas whose evaluation is to be approximated by the techniques discussed in the text. The areas under the

curve can for example be approximated by rectangular boxes or trapezoids.

• The strategy then is to find a reliable polynomial approximation forf(x)in the various
  intervals. Choosing a given approximation forf(x), we obtain a specific approximation to
  the integral.


• With this approximation tof(x)we perform the integration by computing the integrals over
  all subintervals.
  Such a small measure may seemingly allow for the derivation of various integrals. To see this,
  we rewrite the integral as
  ∫b
  a

f(x)dx=

∫a+ 2 h

a

f(x)dx+

∫a+ 4 h

a+ 2 h

f(x)dx+...

∫b

b− 2 h

f(x)dx.

One possible strategy then is to find a reliable polynomial expansion forf(x)in the smaller

subintervals. Consider for example evaluating

∫a+ 2 h

a

f(x)dx,

which we rewrite as ∫a+ 2 h

a

f(x)dx=

∫x 0 +h

x 0 −h

f(x)dx. (5.2)

We have chosen a midpointx 0 and have definedx 0 =a+h. Using Lagrange’s interpolation

formula from Eq. (3.9), an equation we restate here,