
Integration 11-11
1. f x dx
a
a
( ) .
∫
= 0
Proof: We have by the definition of definite integral
f x dx f x x x
b a
n
a
b
n
i
n
i
( ) lim ( ) , .
*
∫
=
∑
=
−
→∞
=
∆ ∆
1
where
Thus,
f x dx f x x
f x x
a a
a
a
n
i
n
n
i
n
i
i
( ) lim ( )
lim ( )( )
*
*
∫
=
∑
=
∑
=
−
→∞
=
→∞
=
∆
∆
1
1
0
nn
n
=
=
=
→∞
0
0
0
lim ( )
.
2.
f x dx f x dx
a
b
b
a
( ) ( )
∫
= −
∫
Proof: We have by the definition of definite integral
f x dx f x x x
b a
n
a
b
n
i
n
i
( ) lim ( ) , .
*
∫
=
∑
=
−
→∞
=
∆ ∆
1
where
We also have
f x dx f x x x
a b
n
b
a
n
i
n
i
( ) lim ( ) , .
*
∫
=
∑
=
−
→∞
=
∆ ∆
1
where
Hence,
f x dx f x
b a
n
f x
a
a
b
n
i
n
n
i
i
( ) lim ( )
lim ( )
(
*
*
∫
=
−
∑
=
− −
→∞
=
→∞
1
bb
n
f x
a b
n
i
n
n
i
n
i
)
lim ( )
*
∑
= −
−
∑
=
→∞
=
1
1
= −
−
∑
= −
∫