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## Wednesday, 27 March 2013

### Properties of the Definite Integral

The following properties are easy to check:
Theorem. If f (x) and g(x) are defined and continuous on [ab], except maybe at a finite number of points, then we have the following linearity principle for the integral:
(i)
f (x) + g(x)dx = f (x) dx + g(x) dx;
(ii)
f (x) dx = f (x) dx, for any arbitrary number .
The next results are very useful in many problems.
Theorem. If f (x) is defined and continuous on [ab], except maybe at a finite number of points, then we have
(i)
f (x) dx = 0;
(ii)
f (x) dx = f (x) dx + f (x) dx;
(iii)
f (x) dx = - f (x) dx;
for any arbitrary numbers a and b, and any c  [ab].The property (ii) can be easily illustrated by the following picture:

Remark. It is easy to see from the definition of lower and upper sums that if f (x) is positive then f (x) dx  0. This implies the following

If f (x g(x) for x  [ab]         f (x) dx  g(x) dx  .

Example. We have

(x2 - 2x)dx = x2 dx - 2x dx  .

We have seen previously that
x2 dx =     and    x dx =    .

Hence
(x2 - 2x)dx =  - 2 = -    .