Slides: C2 - Chapter 11 - Integration

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C2 Chapter 11 Integration
Dr J Frost ([email protected])
Last modified: 17th October 2013
Recap
2 2
2 3 3 2
+ 3  =  +  + ?

3
2
1++

7
−3
2
+
3
1 2 1 3 1 4
 =  +  +  ? +  + 
2
3
4
3 −4
 = −  3 + 
4
?
1
2+ 
−2
−1
 = −2 − 2 ?+ 
2

 
2 −3
 = −  2 + 
4

3
?
Definite Integration

Suppose you wanted to find
the area under the curve
between  =  and  = .




We could add together the area of individual strips,
which we want to make as thin as possible…
Definite Integration

 = ()

1

2
3 4 5 6

7

What is the total area between 1 and 7 ?
7

  
=1
  
As  → 0

Definite Integration

  

You could think of this as “Sum the values of () between  =  and  = .”

Reflecting on above, do you think the
following definite integrals would be
positive or negative or 0?
 = sin 

2
−

+

0
sin  
−

+
0

sin  
−

+
0
sin  
0

2

2
0
2

2
Evaluating Definite Integrals
2
3 2 
1
=  3? 12
= 23 −? 13
=7
?


 ′   =  


We use square brackets to
say that we’ve integrated the
function, but we’re yet to
involve the limits 1 and 2.
Then we find the difference
when we sub in our limits.
=   − ()
Evaluating Definite Integrals
2
2 3 + 2 
1
1 4
=  + 2
2
2
1
1
= 8 + 4 −? + 1
2
21
=
2
−1
4 3 + 3 2 
−2
=  4 +  3 −1
−2
= 1 − 1 −? 16 − 8
= −8
Bro Tip: Be careful with your
negatives, and use
bracketing to avoid errors.
Exercise 11B
1
Find the area between the curve with equation  =   the -axis and the lines
 =  and  = .
a   = 3 2 − 2 + 2
c   =  + 2
8
e   = 3 + 
2
 = 0,  = 2
 = 1,  = 2
 = 1,  = 4
?
?
?

. 


The sketch shows the curve with equation y = ( 2 − 4). Find the area of the
shaded region (hint: first find the roots).

?
4
6
Find the area of the finite region between the curve with equation
 = (3 − )(1 + ) and the -axis.

?

Find the area of the finite region between the curve with equation  =  2 2 − 
and the -axis.

?

Harder Examples
Find the area bounded between the curve with equation  =  3 −
 and the -axis.

Sketch:
(Hint: factorise!)
?
−1
1

1
Looking at the sketch, what is −1  3 −   and why?
0, because the positive and negative
? region cancel each other out.
What therefore should we do?
Find the negative and positive region separately.
0
1 3
1
3 − 3  = − 1


−
3

=
+
−1
0


So total area is +
4


=


?
4
Harder Examples
Sketch the curve with equation  =   − 1  + 3 and find the area
between the curve and the -axis.
The Sketch
The number crunching
  − 1  + 3 =  3 − 2 2 − 3

0
−3
1
?
-3
1

0
 3 − 2 2 − 3  = 11.25
3
−
2 2
7
− 3  = −
?
12
7
5
Adding: 11.25 + 12 = 11 6
Exercise 11C
Find the area of the finite region or regions bounded by the
curves and the -axis.
1
 = +2
2
 = +1 −4
3
 = +3  −3
2
4
=
−2
5
 = −2 −5
1
1?
3
5
20
?6
1
40?
2
1
1?
3
1
21?
12
Curves bound between two lines
 = ()





Remember that  () meant the sum of all the  values
between  =  and  =  (by using infinitely thin strips).
Curves bound between two lines



How could we use a similar principle if we were looking for the
area bound between two lines?
What is the height of each of these strips?
  −
? ()

therefore
area…
=

  ?−  
Curves bound between two lines
Find the area bound between
 =  and  =  4 −  .

3
 4− ?
−   = 4.5
0

Bro Tip: Always do the function of the top
line minus the function of the bottom line.
That way the difference in the  values is
always positive, and you don’t have to
worry about negative areas.
Bro Tip: We’ll need to
find the points at
which they intersect.
Curves bound between two lines
Edexcel C2 May
2013 (Retracted)
 = −4,
 =?2
Area ?
= 36
More complex areas
Bro Tip: Sometimes we can
subtract areas from others.
e.g. Here we could start with
the area of the triangle OBC.
C
A
B

 = ?

Exercise 11D
1 A region is bounded by the line  = 6 and the curve  =  2 + 2.
a) Find the coordinates of the points of intersection.
b) Hence find the area of the finite region bounded by  and the curve.
3 The diagram shows a sketch of part of the curve with equation
2
3
 = 9 − 3 − 5 −  and the line with equation  = 4 − 4.
The line cuts the curve at the points  −1,8 and  1,0 .
Find the area of the shaded region between  and the curve.
 −2,6  2,6
2
 = 10
3
?


2
3
?
6
4
Find the area of the finite region bounded by the curve with equation  =
1 −   + 3 and the line  =  + 3.
4.5
?
9 The diagram shows part of the curve with equation  = 3  −  3 + 4 and the line
1
with equation  = 4 − 2 .
a) Verify that the line and the curve cross at  4,2 .
b) Find the area of the finite region bounded by the curve and the line. 4
?
7.2

Exercise 11D
(Probably more difficult than you’d see in an exam paper, but you never know…)
Q6
The diagram shows a sketch of part of the curve with equation  =
 2 + 1 and the line with equation  = 7 − .
a) Find the area of 1 .
b) Find the area of 2 .

7
1
5
6
1
2 = 17
6
1 = 20
?
2
7

Trapezium Rule
y4
y3
What is the area here?
y2

1
= ℎ 1 + 2
2
1 ?
+ ℎ 2 + 3
2
1
+ ℎ 3 + 4
2
y1
h
Instead of infinitely thin
rectangular strips, we
might use trapeziums to
approximate the area
under the curve.
h
h
Trapezium Rule
In general:
width of each trapezium


ℎ
  ≈ 1 + 2 2 + ⋯ + −1 + 
2
Area under curve
is approximately
Example
We’re approximating the region bounded between  = 1,
 = 3, the x-axis the curve  =  2
x
1
1.5
2
2.5
3
y
1
2.25
4
6.25
9
?
ℎ = 0.5
?
 ≈ 8.75
Trapezium Rule
May 2013 (Retracted)
Bro Tip: You can generate table with Casio calcs .  → 3 (). Use ‘Alpha’ button to key in X within the function. Press =
0.8571
?
 =
. 
.  +  .  + .  + .
?  + .  + .  = . 

To add: When do we underestimate and overestimate?

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