C2 Chapter 11: Integration
Dr J Frost (jfrost@tiffin.kingston.sch.uk)
www.drfrostmaths.com
Last modified: 1st September 2015
Recap
2𝑥 2
2 3 3 2
+ 3𝑥 𝑑𝑥 = 𝑥 + 𝑥 + ?
𝑐
3
2
1+𝑥+
𝑥
7
−3
𝑥2
+
𝑥3
1 2 1 3 1 4
𝑑𝑥 = 𝑥 + 𝑥 + 𝑥 ? + 𝑥 + 𝑐
2
3
4
3 −4
𝑑𝑥 = − 𝑥 3 + 𝑐
4
?
1
2+ 𝑥
−2
−1
𝑑𝑥 = −2𝑥 − 2𝑥 ?+ 𝑐
2
𝑥
𝑥 𝑥
2 −3
𝑑𝑥 = − 𝑥 2 + 𝑐
4
𝑥
3
?
Definite Integration
𝑦
Suppose you wanted to find
the area under the curve
between 𝑥 = 𝑎 and 𝑥 = 𝑏.
𝑥
𝑎
𝛿𝑥
𝑏
We could add together the area of individual strips,
which we want to make as thin as possible…
Definite Integration
𝑦
𝑦 = 𝑓(𝑥)
𝛿𝑥
𝑥1
𝑎
𝑥2
𝑥3 𝑥4 𝑥5 𝑥6
𝑥
𝑥7
𝑏
What is the total area between 𝑥1 and 𝑥7 ?
𝑏
7
𝑓 𝑥𝑖 𝛿𝑥
𝑖=1
𝑓 𝑥 𝑑𝑥
As 𝛿𝑥 → 0
𝑎
Definite Integration
𝑏
𝑓 𝑥 𝑑𝑥
𝑎
You could think of this as “Sum the values of 𝑓(𝑥) between 𝑥 = 𝑎 and 𝑥 = 𝑏.”
𝑦
Reflecting on above, do you think the
following definite integrals would be
positive or negative or 0?
𝑦 = sin 𝑥
𝜋
2
−

+

0
sin 𝑥 𝑑𝑥
−

+
0

sin 𝑥 𝑑𝑥
−

+
0
sin 𝑥 𝑑𝑥
0
𝜋
2𝜋
𝑥
2𝜋
0
2𝜋
𝜋
2
Evaluating Definite Integrals
2
3𝑥 2 𝑑𝑥
1
= 𝑥 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.
𝑏
𝑓 ′ 𝑥 𝑑𝑥 = 𝑓 𝑥
𝑎
𝑏
𝑎
= 𝑓 𝑏 − 𝑓(𝑎)
Evaluating Definite Integrals
−1
2
4𝑥 3 + 3𝑥 2 𝑑𝑥
2𝑥 3 + 2𝑥 𝑑𝑥
1
1 4
= 𝑥 + 𝑥2
2
2
1
1
= 8 + 4 −? + 1
2
21
=
2
−2
= 𝑥 4 + 𝑥 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 𝑦 = 𝑓 𝑥 the 𝑥-axis and the lines
𝑥 = 𝑎 and 𝑥 = 𝑏.
a 𝑓 𝑥 = 3𝑥 2 − 2𝑥 + 2
c 𝑓 𝑥 = 𝑥 + 2𝑥
8
e 𝑓 𝑥 = 𝑥3 + 𝑥
2
𝑎 = 0, 𝑏 = 2
𝑎 = 1, 𝑏 = 2
𝑎 = 1, 𝑏 = 4
?
?
?
𝟖
𝟒. 𝟐𝟐
𝟏𝟎𝟏
𝟏𝟐
The sketch shows the curve with equation y = 𝑥(𝑥 2 − 4). Find the area of the
shaded region (hint: first find the roots).
𝟒
?
𝟐
𝟏𝟎?
𝟑
4
Find the area of the finite region between the curve with equation
𝑦 = (3 − 𝑥)(1 + 𝑥) and the 𝑥-axis.
6
Find the area of the finite region between the curve with equation 𝑦 = 𝑥 2 2 − 𝑥
and the 𝑥-axis.
𝟏
𝟏?
𝟑
Harder Examples
Find the area bounded between the curve with equation 𝑦 = 𝑥 3 −
𝑥 and the 𝑥-axis.
𝑦
Sketch:
(Hint: factorise!)
?
1
−1
𝑥
1
Looking at the sketch, what is −1 𝑥 3 − 𝑥 𝑑𝑥 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 𝑑𝑥 = − 1
𝑥
𝑥
−
3
𝑑𝑥
=
+
−1
0
𝟏
𝟒
So total area is +
4
𝟏
𝟏
=
𝟒
𝟐
?
4
Harder Examples
Sketch the curve with equation 𝑦 = 𝑥 𝑥 − 1 𝑥 + 3 and find the area
between the curve and the 𝑥-axis.
The Sketch
The number crunching
𝑥 𝑥 − 1 𝑥 + 3 = 𝑥 3 − 2𝑥 2 − 3𝑥
𝑦
0
𝑥 3 − 2𝑥 2 − 3𝑥 𝑑𝑥 = 11.25
−3
1
𝑥3
?
-3
1
𝑥
0
−
2𝑥 2
7
− 3𝑥 𝑑𝑥 = −
?
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 𝑥-axis.
1
𝑦 =𝑥 𝑥+2
2
𝑦 = 𝑥+1 𝑥−4
3
𝑦 = 𝑥+3 𝑥 𝑥−3
𝑥2
4
𝑦=
𝑥−2
5
𝑦 =𝑥 𝑥−2 𝑥−5
1
1?
3
5
20
?6
1
40?
2
1
1?
3
1
21?
12
Curves bound between two lines
𝑦 = 𝑓(𝑥)
𝛿𝑥
𝑥
𝑎
𝒃
𝑏
Remember that 𝒂 𝒇(𝒙) meant the sum of all the 𝑦 values
between 𝑥 = 𝑎 and 𝑥 = 𝑏 (by using infinitely thin strips).
Curves bound between two lines
𝑏
𝑎
𝑥
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?
𝑔 𝑥 −
? 𝑓(𝑥)
𝑏
therefore
area…
𝐴=
𝑎
𝑔 𝑥 ?− 𝑓 𝑥
Curves bound between two lines
Find the area bound between
𝑦 = 𝑥 and 𝑦 = 𝑥 4 − 𝑥 .
𝑦
3
𝑥 4−𝑥 ?
− 𝑥 𝑑𝑥 = 4.5
0
𝑥
Bro Tip: Always do the function of the top
line minus the function of the bottom line.
That way the difference in the 𝑦 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)
𝑥 = −4,
𝑥 =?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
𝟏
𝑨𝒓𝒆𝒂 = 𝟏𝟔?
𝟑
Exercise 11D
1 A region is bounded by the line 𝑦 = 6 and the curve 𝑦 = 𝑥 2 + 2.
a) Find the coordinates of the points of intersection.
b) Hence find the area of the finite region bounded by 𝐴𝐵 and the curve.
3 The diagram shows a sketch of part of the curve with equation
2
3
𝑦 = 9 − 3𝑥 − 5𝑥 − 𝑥 and the line with equation 𝑦 = 4 − 4𝑥.
The line cuts the curve at the points 𝐴 −1,8 and 𝐵 1,0 .
Find the area of the shaded region between 𝐴𝐵 and the curve.
𝐴 −2,6 𝐵 2,6
2
𝐴𝑟𝑒𝑎 = 10
3
?
𝐴
𝐵
2
3
?
6
4
Find the area of the finite region bounded by the curve with equation 𝑦 =
1 − 𝑥 𝑥 + 3 and the line 𝑦 = 𝑥 + 3.
4.5
?
9 The diagram shows part of the curve with equation 𝑦 = 3 𝑥 − 𝑥 3 + 4 and the line
1
with equation 𝑦 = 4 − 2 𝑥.
a) Verify that the line and the curve cross at 𝐴 4,2 .
b) Find the area of the finite region bounded by the curve and the line. 4
?
7.2
𝐴
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 𝑦 =
𝑥 2 + 1 and the line with equation 𝑦 = 7 − 𝑥.
a) Find the area of 𝑅1 .
b) Find the area of 𝑅2 .
𝑦
7
𝑅1
5
6
1
𝑅2 = 17
6
𝑅1 = 20
?
𝑅2
7
𝑥
Trapezium Rule
y4
y3
Instead of infinitely thin
rectangular strips, we
might use trapeziums to
approximate the area
under the curve.
What is the area here?
y2
𝐴𝑟𝑒𝑎
1
= ℎ 𝑦1 + 𝑦2
2
1 ?
+ ℎ 𝑦2 + 𝑦3
2
1
+ ℎ 𝑦3 + 𝑦4
2
y1
h
h
h
Trapezium Rule
In general:
width of each trapezium
𝑏
𝑎
ℎ
𝑦 𝑑𝑥 ≈ 𝑦1 + 2 𝑦2 + ⋯ + 𝑦𝑛−1 + 𝑦𝑛
2
Area under curve
is approximately
Example
We’re approximating the region bounded between 𝑥 = 1,
𝑥 = 3, the x-axis the curve 𝑦 = 𝑥 2
x
1
1.5
2
2.5
3
y
1
2.25
4
6.25
9
?
ℎ = 0.5
?
𝐴𝑟𝑒𝑎 ≈ 8.75
Trapezium Rule
May 2013 (Retracted)
Bro Tip: You can generate table with Casio calcs . 𝑀𝑜𝑑𝑒 → 3 (𝑇𝑎𝑏𝑙𝑒). Use ‘Alpha’ button to key in X within the function. Press =
0.8571
?
𝑨𝒓𝒆𝒂 =
𝟎. 𝟏
𝟎. 𝟕𝟎𝟕𝟏 + 𝟐 𝟎. 𝟕𝟓𝟗𝟏 + 𝟎. 𝟖𝟎𝟗𝟎 + 𝟎.
? 𝟖𝟓𝟕𝟏 + 𝟎. 𝟗𝟎𝟑𝟕 + 𝟎. 𝟗𝟒𝟖𝟕 = 𝟎. 𝟒𝟏𝟔
𝟐
To add: When do we underestimate and overestimate?