Chapter 6
Application of Integration
6.1 Riemann Sum and Definite Integral
Main Properties
1)
( ) =− ( ) .
2)
( ) = 0.
3)
( ) = ( ) ℎ .
4)
[ ( ) ± ( )] = ( ) ± ( ) .
5)
( ) = ( ) + ( ) ℎ .
6)
( ) = ( ) .
7)
= ( − ).
8) If ( ) ≥ 0 for ≤ ≤ then
( ) ≥ 0.
9) If ( ) ≥ ( ) for ≤ ≤ then
( ) ≥ ( ) .
10) If ≤ ( )≤ for ≤ ≤ then
( − )≤ ( ) ≤ ( − ).
Mathematical Analysis 84
, 11)
( ) ≤ | ( )|
12) If ( ) is an even function then,
( ) =2 ( ) .
Likewise, if ( ) is an odd function then,
( ) =0
Example 6.1 Evaluate
) √ ) ( + ) ) 1−
Solution
/
2 2
) √ = − √ =− =− − (0) = − .
3/2 3 3
1 1 5
) ( + ) = + = + − (0 + 0) = .
2 3 2 3 6
c) Let = sin ⟹ = cos
=0⟹ =0 =1⟹ =
2
1
1− = cos cos = cos = (1 + cos 2 )
2
1 1 1 1 1
= + sin 2 = + sin − 0 + sin 0 =
2 2 2 2 2 2 4
6.2 Area under Graph of Function
Let be a function that is continuous on a closed and bounded interval [ , ].
Suppose that is nonnegative on [ , ].
Mathematical Analysis 85
Application of Integration
6.1 Riemann Sum and Definite Integral
Main Properties
1)
( ) =− ( ) .
2)
( ) = 0.
3)
( ) = ( ) ℎ .
4)
[ ( ) ± ( )] = ( ) ± ( ) .
5)
( ) = ( ) + ( ) ℎ .
6)
( ) = ( ) .
7)
= ( − ).
8) If ( ) ≥ 0 for ≤ ≤ then
( ) ≥ 0.
9) If ( ) ≥ ( ) for ≤ ≤ then
( ) ≥ ( ) .
10) If ≤ ( )≤ for ≤ ≤ then
( − )≤ ( ) ≤ ( − ).
Mathematical Analysis 84
, 11)
( ) ≤ | ( )|
12) If ( ) is an even function then,
( ) =2 ( ) .
Likewise, if ( ) is an odd function then,
( ) =0
Example 6.1 Evaluate
) √ ) ( + ) ) 1−
Solution
/
2 2
) √ = − √ =− =− − (0) = − .
3/2 3 3
1 1 5
) ( + ) = + = + − (0 + 0) = .
2 3 2 3 6
c) Let = sin ⟹ = cos
=0⟹ =0 =1⟹ =
2
1
1− = cos cos = cos = (1 + cos 2 )
2
1 1 1 1 1
= + sin 2 = + sin − 0 + sin 0 =
2 2 2 2 2 2 4
6.2 Area under Graph of Function
Let be a function that is continuous on a closed and bounded interval [ , ].
Suppose that is nonnegative on [ , ].
Mathematical Analysis 85
