“Teach A Level Maths”
Vol. 1: AS Core Modules
25: Definite Integration
© Christine Crisp
Definite Integration
Module C1
AQA
Module C2
Edexcel
MEI/OCR
OCR
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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
Definite Integration
Definite Integration
The following slides contain repeats of
information on earlier slides, shown without
colour, so that they can be printed and
photocopied.
For most purposes the slides can be printed
as “Handouts” with up to 6 slides per sheet.
Definite Integration
Evaluating the Definite Integral
The definite integration results in a value.
e.g.1
1
2 3 x
2
 2 dx






1
So,

2


1
x
12
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