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IBSL Maths
Definite Integration
© Christine Crisp
Definite Integration
e.g.1
1
2
3 x 2  2 dx
is a definite integral
The numbers on the integral sign are called
the limits of integration
Definite Integration
Evaluating the Definite Integral
The definite integration results in a value.
e.g.1
1
2
3 x 2  2 dx

1
3x 3
 2x
31
 Find the indefinite integral but omit C
Definite Integration
Evaluating the Definite Integral
The definite integration results in a value.
e.g.1
1
2 3 x
2
 2 dx



1
x
3

 2x 
 2
 Draw square brackets and hang the
limits on the end
Definite Integration
Evaluating the Definite Integral
The definite integration results in a value.
e.g.1
1
2 3 x
2
 2 dx

1






12
 Replace x with •
•
x
3

 2x 
 2




3
(

2
)



 2(2) 
the top limit
the bottom limit

Definite Integration
Evaluating the Definite Integral
The definite integration results in a value.
e.g.1
1
2 3 x
2
 2 dx

1






12
x
3

 2x 
 2





3
(

2
)


 Subtract and evaluate

 2(2) 

Definite Integration
Evaluating the Definite Integral
The definite integration results in a value.
e.g.1
1
2 3 x
2
 2 dx







12

1
So,

2
1
x
3

 2x 
 2




3   84 
 15
2

3
(

2
)


3 x  2 dx  15

 2(2) 

Definite Integration
SUMMARY
The method for evaluating the definite integral is:
 Find the indefinite integral but omit C
 Draw square brackets and hang the
limits on the end
 Replace x with •
•
the top limit
the bottom limit
 Subtract and evaluate
Definite Integration
Evaluating the Definite Integral
1
1
e.g. 2 Find
x

7x
x  7 x  12 dx  

 12 x 
2
 3

1
1
Solution:
x 2  7 x  12 dx

2
3
2
Indefinite integral but no C
1
1
Definite Integration
Evaluating the Definite Integral
1
1
e.g. 2 Find
x

7x
x  7 x  12 dx  

 12 x 
2
 3

1
1
Solution:
x 2  7 x  12 dx

2
3
2
1
1
3
2


1
7
(

1
)
7
(

1
)


Substitute
for
x
:
    12   

 12( 1) 
top
minus
limit
2
3
2
 3 limit

 bottom

1 7

 1 7

    12      12
Simplify
3 2

 3 2

Definite Integration
Evaluating the Definite Integral
1 7

    12
3 2

 1 7

  3  2  12



In this example, if we can’t use a calculator,
we can
We
must
be
1
7
1
7
save time by collecting
  12terms
 from
 12both brackets.
very careful
3
2
2
  24
3
2
 24
3
3
2
with the signs
7
7


0
2
2
Definite Integration
Exercises
2
1
1. Find
1
2
3 x

4x
3 x  4 x  1 dx  

 x
21
 31
1
2
3
2
  ( 2) 3  2( 2) 2  2    1  2  1 

 

2
 14  2  12
1
2

6
x
2
x
2. Find
6 x  2 x  3 dx  

 3 x
21
 31
2
2

2
2
2
3
2
  2( 2) 3  1( 2) 2  3( 2)    2( 2) 3  ( 2) 2  3( 2) 

 

  16  4  6     16  4  6   6  ( 14)  20