Page 533 - Modelling in Transport Phenomena A Conceptual Approach

A.8. METHODS OF IlVTEGRATION 513

If f(x) is a monotonic function, then the value of I is bounded by Imin and
I,, such that

Monotonically increasing function Imin = f(a)(b-a)

f (3) = (A. 8-3)
Monotonically decreasing function Imin = f(b)(b-a)
Imm = f (a)(b - a)
In some cases, only part of the integrand may be approximated to permit analytical
integration, i.e.,

I = J,” f (x)g(x) dx = (A.8-4)

Example A.5 Evaluate the integral

Solution

Analytical evaluation of the integral is possible and the result is

I = loxzdmdx

- 2 (0.15 x2 - 2.4s + 32)
- dmlX=lo 552.4
=
0.105 x=o
The same integml can be evaluated appmximately as follows: Note that the inte-
grand is the product of two tern and the integral can be wdten as

where
f(x) = x2 and g(x) = 4- (2)

The value of g(x) is 1.732 and 1.414 at x = 10 and x = 0, respectively. Since the
value of g(x) does not change dwtically over the interval 0 5 x 5 10, Eq. (1)
can be expressed in the form