Page 534 - Modelling in Transport Phenomena A Conceptual Approach

514 APPENDIX A. MATHEMATICfi PRELIMINARIES

As a rough appmximation, the average wdue of the function g, (g), can be taken
as the arithmetic average, i.e.,

1.732 + 1.414 = 1.573
(9) = 2 (4)

Therefore, Eq. (3) becomes

I = 1.5731 x2dx = - = 524.3 (5)
3 z=o

with a percent error of approximately 5%.

A.8.2 Graphical Integration

In order to evaluate the integral given by Eq. (A.8-1) graphically, first f(x) is
plotted as a function of 2. Then, the area under this curve in the interval [a, b] is
determined.

A.8.3 Numerical Integration or Quadrature

Numerical integration or quadrature2 is an alternative to graphical and analytical
integration. In this method, the integrand is replaced with a polynomial and this
polynomial is integrated to give a summation:

(A.&5)

Numerical integration is preferred for the following cases:

0 The function f(x) is not known but the values of f(x) is known at equally
spaced discrete points.

0 The function f(z) is known, but too difficult to integrate analytically.

A.8.3.1 Numerical integration with equally spaced base points

Consider Figure A.5 in which f(x) is known only at 5 equally spaced base points.
The two most frequently used numerical integration methods for this case are the
trapezoidal rule and the Simpson’s rule.

2The word quadrature is used for approximate integration.