Lecture 16 : Integral as a limit of Riemann sums [Section 16.2]
16.2.5
Theorem (Properties of Integral):
Let be bounded functions.
(i) If is integrable and , then
(ii) If is integrable, then is integrable and
(iii) If is integrable and , then is integrable and .
(iv) If and are integrable, then are integrable and
(v) If and are integrable and for all , then
(vi) If is integrable on , then is integrable over every interval and
This is called the additive property of the integral.
(we define for every ).
(vii)If and are integrable, then is also integrable.
Proof:
Proofs of all these properties follow from the properties of limits. Though not difficult, the proofs are technical. We shall assume them. Interested reader can refer a book on Real Analysis. Back
be bounded functions. 
is integrable and 
, then 
is integrable, then 
is integrable and
is integrable and 
, then 
is integrable and 
.
and 
are integrable, then 
are integrable and 
and 
are integrable and 
for all 
, then 
is integrable on 
, then 
is integrable over every interval 
and 
for every 
). 
and 
are integrable, then 
is also integrable. 
