Properties of the Definite Integral
The following properties are easy to check:
Theorem. If f (x) and g(x) are defined and continuous
on [a, b], 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 [a, b],
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 [a, b].
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 [
a,
b]
f (
x) d
x g(
x) d
x .
Example. We have
(
x^{2}  2
x)
dx =
x^{2} d
x  2
x d
x .
We have seen previously that
x^{2} d
x =
and
x dx =
^{ . }
Hence
(
x^{2}  2
x)
dx =
 2
= 
^{ . }
Exercise 1. Given that
f (
x)
dx = 2 ,
f (
x)
dx =  1 ,
find
f (x)dx.
Answer.
Exercise 2. Let f (x) be defined and continuous on [a, b].
Assume that f (x) is positive. Show that the function
F(
x) =
f (
t)
dt
is increasing on [a, b].
Answer.
Exercise 3. Let f (x) and g(x) be two functions defined
and continuous on [a, b]. Show that
f (
x)
g(
x)
dx f (
x)
dx^{ . }g(
x)
dx .
Answer.
For more on the Area Problem, click HERE.
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