THE DEFINITE INTEGRAL PROPERTIES OF THE DEFINITE INTEGRAL In this section, we will discuss the basic properties of the definite integral. We will then move on to illustrate each property with a number of examples.
Property 1 When we defined the definite integral
∫
b
f(x) dx
a
we did so with the understanding that a < b (since a is the lower limit and b is the upper limit). On the contrary, the definite integral is also defined even when a > b. Recall that
∆x
=
(b – a)/n
(if a < b)
But, if a > b, then ∆x will be NEGATIVE. Therefore, if a > b, then
∫
b a
n
f(x) dx
∑ – i=1
=
f(x*i ) ∆x
=
–∫
a b
f(x) dx
This leads us to Property 1 of definite integrals:
∫
b a
– ∫
=
f(x) dx
a b
f(x) dx
(if a > b)
Property 2 If a = b, then
∆x
=
(b – a)/n
∆x
=
0/n
∆x
=
0
Therefore,
∫
b a
n
f(x) dx
∑
=
i=1
f(x*i ) ∆x
and this gives
n
n
∑ i=1
f(x i ) ∆x *
∑
=
i=1
f(x*i ) 0
=
0
This leads us to property 2 of definite integrals:
∫
b a
f(x) dx
=
0
(if a = b)