1. P1 Chapter 13 :: Integration
jfrost@tiffin.kingston.sch.uk
www.drfrostmaths.com
@DrFrostMaths
Last modified: 12th March 2020
2. Use of DrFrostMaths for practice
Register for free at:
www.drfrostmaths.com/homework
Practise questions by chapter, including
past paper Edexcel questions and extension
questions (e.g. MAT).
Teachers: you can create student accounts
(or students can register themselves).
3. Chapter Overview
1:: Find π¦ given
ππ¦
ππ₯ 2:: Evaluate definite integrals, and hence
the area under a curve.
This chapter is roughly divided into two parts: the first, indefinite integration, is the opposite of
differentiation. The second, definite integration, allows us to find areas under graphs (as well as
surface areas and volumes) or areas between two graphs.
Find the area bounded between the curve
with equation π¦ = π₯2 β 2π₯ and the π₯-axis.
A curve has the gradient
function
ππ¦
ππ₯
= 3π₯ + 1
If the curve goes through the
point 2,3 , determine π¦.
3:: Find areas bound between two
different lines.
Find the points of intersection of
π¦ = π₯2 β 4π₯ + 3 and π¦ = π₯ + 9,
and hence find the area bound
between the two lines.
4. Integrating π₯π
terms
Integration is the opposite of differentiation.
(For this reason it is also called βantidifferentiationβ)
5π₯3
multiply by power reduce power by 1
15π₯2
increase power by 1
divide by new power
Find π¦ when
ππ¦
ππ₯
= 3π₯2
Adding 1 to the power and dividing by this power give us:
π¦ = π₯3
However, other functions would also have differentiated to 3π₯2
:
π¦ = π₯3
+ 1, π¦ = π₯3
β 4, β¦
Clearly we could have had any constant, as it disappears upon
differentiation.
β΄ π = ππ + π
However, thereβs one added complicationβ¦
Function
Gradient Function
π is known as a constant of
integration
?
? ?
5. Textbook Error on Pg289: βWhen integrating
polynomialsβ, but Example 3 is not a polynomial
because it has negative and fractional powers.
Examples
Find π¦ when:
ππ¦
ππ₯
= 4π₯3
π¦ = π₯4
+ π
ππ¦
ππ₯
= π₯5 π¦ =
1
6
π₯6 + π
Exam Note: Historically βC1β
penalised the lack of +π, but
modules thereafter didnβt. You
should always include it.
You could also write as
π₯6
6
. Itβs a
matter of personal preference.
ππ¦
ππ₯
= 3π₯
1
2 π¦ = 3
2
3
π₯
3
2 + π = 2π₯
3
2 + π
Fro Tip: Many students are taught to write
3π₯
3
2
3
2
(as does textbook!). This is ugly and students
then often struggle to simplify it. Instead
remember back to Year 7: When you divide by
a fraction, you multiply by the reciprocal.
?
?
?
6. More Fractional Examples
Find π¦ when:
ππ¦
ππ₯
=
4
π₯
= 4π₯β
1
2 π¦ = 8π₯
1
2 + π
When we divide by
1
2
we
multiply by the reciprocal, i.e. 2.
Fro Tip: I recommend doing in
your head when the
simplification would be simple.
ππ¦
ππ₯
= 5π₯β2 π¦ = β5π₯β1 + π
ππ¦
ππ₯
= 4π₯
2
3 π¦ = 4
3
5
π₯
5
3 + π =
12
5
π₯
5
3 + π
ππ¦
ππ₯
= 10π₯β
2
7 π¦ = 10
7
5
π₯
5
7 + π = 14π₯
5
7 + π
? ?
?
?
?
7. Test Your Understanding
Find π(π₯) when:
πβ²
π₯ =
2
π₯7
= 2π₯β7
π π₯ = β
1
3
π₯β6
+ π
πβ² π₯ = 3
π₯ = π₯
1
3 π π₯ =
3
4
π₯
4
3 + π
πβ² π₯ = 33π₯
5
6 π π₯ = 18π₯
11
6 + π
? ?
? ?
πβ² π₯ = 2π₯ + 7 π π₯ = π₯2 + 7π₯ + π
πβ² π₯ = π₯2 β 1 π π₯ =
1
3
π₯3 β π₯ + π
?
?
Note: In case youβre wondering what happens if
ππ¦
ππ₯
=
1
π₯
= π₯β1
, the problem is that after
adding 1 to the power, weβd be dividing by 0. You will learn how to integrate
1
π₯
in Year 2.
?
9. Integration notation
The following notation could be used to differentiate an expression:
π
ππ₯
5π₯2 = 10π₯
There is similarly notation for integrating an expression:
β« 10π₯ ππ₯ = 5π₯2
+ π
The ππ₯ here means
differentiating βwith
respect to π₯β.
βIntegrateβ¦β
ββ¦this expressionβ
ββ¦with respect to π₯β
(the ππ₯ is needed just as it was needed in the
differentiation notation at the top of this slide)
This is known as indefinite integration, in contrast to definite integration, which
weβll see later in the chapter.
It is called βindefiniteβ because the exact expression is unknown (due to the +π).
10. Examples
Find β«(π₯β
3
2 + 2) ππ₯
Fro Note: The brackets are required if
thereβs multiple terms.
= β2π₯β
1
2 + 2π₯ + π
Find β«(6π‘2 β 1) ππ‘
= 2π‘3
β π‘ + π
Note the ππ‘ instead of ππ₯.
Find β«(ππ₯3 + π) ππ₯ where π and π are constants.
=
1
4
ππ₯4 + ππ₯ + π
Textbook (Minor) Error: βany other letters must be treated as
constantsβ. Similar to the error in the differentiation chapter, it
should read βany other letters, which are either constants or
variables independent of π₯, can be treated as numbersβ. In β« π₯π¦ ππ₯,
if π¦ is a variable, we can only treat π¦ as a constant if it is not
dependent on π₯, i.e. there is not some equation relating π¦ to π₯.
?
?
?
11. Test Your Understanding
Edexcel C1 May 2014(R) Q4b
π¦ = 2π₯5 + 6π₯β
1
2
π¦ ππ₯ =
1
3
π₯6
+ 12π₯
1
2 + π
?
13. Finding constant of integration
Recall that when we integrate, we get a constant of integration, which could be
any real value. This means we donβt know what the exact original function was.
π₯
π¦
πβ²
π₯ = 3π₯2
π₯
π¦
π π₯ = π₯3
π π₯ = π₯3
+ π
π π₯ = π₯3
β 1
β«
?
1,3
But if we know one point
on the curve, it leaves
only one possibility.
The curve with equation π¦ = π(π₯) passes
through 1,3 . Given that πβ² π₯ = 3π₯2,
find the equation of the curve.
πβ² π₯ = 3π₯2
β΄ π π₯ = π₯3 + π
Using the point (1,3): 3 = 13
+ π β΄ π = 2
π π₯ = π₯3
+ 2
?
14. Test Your Understanding
Edexcel C1 May 2014 Q10
To keep you occupied if
you finish (a) quickly!
π¦ β 25 = β
1
2
π₯ β 4
(b)
? a
? b
16. Definite Integration
So far weβve seen integration as βthe opposite of differentiationβ,
allowing us to find π¦ = π π₯ when we know the gradient function
π¦ = πβ²
π₯ .
In practical settings however the most useful use of integration is that it
finds the area under a graph. Remember at GCSE for example when you
estimated the area under a speed-time graph, using trapeziums, to get
the distance?
If you knew the equation of the curve, you could get the exact area!
Before we do this, we need to understand how to find a definite integral:
speed
time
π΄πππ = πππ π‘ππππ
1
5
4π₯3 ππ₯ =
These are known as limits, which give the
values of π₯ weβre finding the area between.
π¦
π₯
π΄πππ = πππ π‘ππππ
1 5
π¦ = 4π₯3
?
We integrate as normal, but put expression in
square brackets, meaning we still need to evaluate
the integrated expression using the limits.
Write β¦ β (β¦ ) and evaluate the expression for
each of the limits, top one first.
π₯4
1
5
= 54
β 14
= 624
17. Another Example
β3
3
π₯2 + 1 ππ₯ =
1
3
π₯3 + π₯
β3
3
=
1
3
3 3 + 3 β
1
3
β3 3 β 3
= 12 β β12
= 24
We DONβT have a
constant of integration
when doing definite
integration. Iβll explain
why later.
Write out you working
EXACTLY as seen here.
The β¦ β (β¦ ) brackets
are particularly crucial
as youβll otherwise likely
make a sign error.
?
?
?
βUse of Technologyβ Monkey says:
You can use the β«π
π
β‘ button on your calculator to
evaluate definite integrals.
But only use it to check your answer.
18. Problem Solving
Given that π is a constant and β«1
5
2ππ₯ + 7 ππ₯ = 4π2
, show that there are two
possible values for π and find these values.
1
5
2ππ₯ + 7 ππ₯ = ππ₯2
+ 7π₯ 1
5
= 25π + 35 β π + 7
= 24π + 28
β΄ 24π + 28 = 4π2
π2 β 6π β 7 = 0
π + 1 π β 7 = 0
π = β1 ππ 7
?
Remember: π is a
constant, so just treat it
as a number.
19. Exercise 13D
Pearson Pure Mathematics Year 1/AS
Page 297
(Classes in a rush may want to skip this exercise and go to the next section, which
continues definite integration, but in the context of areas under graphs).
Extension
[MAT 2009 1A] The smallest value of
πΌ π =
0
1
π₯2
β π 2
ππ₯
as π varies, is what?
π° π =
π
π
ππ
β ππππ
+ ππ
π π
=
π
π
ππ
β
π
π
πππ
+ ππ
π
π
π
=
π
π
β
π
π
π + ππ
We then use differentiation or completing
the square to find that the minimum value
of
π
π
β
π
π
π + ππ
is
π
ππ
.
?
[MAT 2015 1D] Let
π π₯ = β«0
1
π₯π‘ 2
ππ‘ and π π₯ = β«0
π₯
π‘2
ππ‘
Let π΄ > 0. Which of the following
statements are true?
A) π π π΄ is always bigger than π π π΄
B) π π π΄ is always bigger than π π π΄
C) They are always equal.
D) π π π΄ is bigger if π΄ < 1, and π π π΄
is bigger if π΄ > 1.
E) π π π΄ is bigger if π΄ < 1, and π π π΄
is bigger if π΄ > 1.
(Official Sln) Evaluating: π π =
ππ
π
and π π =
ππ
π
. Hence
π π π¨ =
π¨π
ππ and π π π¨ =
π¨π
ππ. Hence answer is (B).
?
1
2
20. Areas under curves
Earlier we saw that the definite integral β«π
π
π(π) π π gives
the area between a positive curve π¦ = π(π₯), the π-axis,
and the lines π₯ = π and π₯ = π.
(Weβll see why this works in a sec)
π¦
π₯
π΄πππ = πππ π‘ππππ
π π
π¦ = π(π₯)
π΄πππ
Find the area of the finite region between the curve
with equation π¦ = 20 β π₯ β π₯2 and the π₯-axis.
π¦
π₯
Factorise in order to find roots:
ππ β π β ππ
= π
ππ
+ π β ππ = π
π + π π β π = π
π = βπ ππ π = π
Therefore area between curve and π-axis is:
βπ
π
ππ β π β ππ
π π = πππ β
π
π
ππ
β
π
π
ππ
βπ
π
= ππ β π β
ππ
π
β βπππ β
ππ
π
+
πππ
π
=
πππ
π
β5 4
?
21. Just for your interestβ¦
Why does integrating a function give you
the area under the graph?
Part 1:
Youβre already familiar with the idea that
gradient is the rate at which a quantity
changes, and we consider an infinitesimally
small change in the variable.
You could consider the gradient as the little
bit youβre adding on each time.
Hereβs some practical examples using
formulae you covered in your younger years!
ππ If you wanted to consider the rate at
which the area of a circle increases
with respect to the radius, consider
a small change ππ in the radius.
The change in area is an infinitely
thin ring, which looks like youβve
drawn a circumference.
So what is this rate?
π΄ = ππ2
β΄
ππ΄
ππ
= 2ππ
OMG THAT IS THE
CIRCUMFERENCE
π
So to draw a shaded circle (which has area!), itβs
a bit like repeatedly drawing the circumference
of a circle with gradually increasing radius.
Since the circumference is what the area is
increasing by each time, the circumference is
the gradient of the area of the circle, and
conversely (by the definition of integration
being the opposite of differentiation), the area
is the integral of the circumference.
You might be rightly upset that you canβt add a length to an area
(only an area to an area, innit).
But by considering the infinitely thin width ππ of the
circumference youβre drawing, it does have area!
ππ΄
ππ
= 2ππ β ππ΄ = 2ππ ππ
i.e. the change in area, ππ΄, is 2ππ Γ ππ
If we βroll outβ the added area weβre adding each time, this forms
a rectangle, whose length hopefully corresponds to the
circumference of the circle:
?
ππ
If the added area is 2ππ ππ and the thickness is ππ, then the
length is
2ππ ππ
ππ
= 2ππ as expected.
22. Since π π₯ = π΄β²(π₯), the area function π΄(π₯) is the
integral of π(π₯). Thus:
π
π
π π₯ ππ₯ = π΄ π₯ π
π
= π΄ π β π΄(π)
Part 2:
It gets better!
Consider the volume of
a sphere:
π =
4
3
ππ3
ππ
ππ
= 4ππ2
OMG THATβS THE
SURFACE AREA
This works for a similar reason to before.
π₯
π(π₯)
π΄(π₯)
And the same principle applies to area under a graph. If π΄(π₯)
gives the area up to π₯, then what weβre βadding on each timeβ
(i.e. the gradient) is sort of like drawing a line of length π(π₯) at
the right-most end each time (or more accurately, an infinitely
thin rectangle of area π π₯ Γ β). Thus if π(π₯) is the gradient of
the area, then conversely the area is the integral of π π₯ .
π₯ + β
This gives us an intuitive sense of why it works, but we need a formal proof:
β
Using the diagram above, the area up to π₯ + β, i.e. π΄(π₯ + β), is
approximately the area up to π₯ plus the added rectangular strip:
π΄ π₯ + β β π΄ π₯ + (π π₯ Γ β)
Rearranging:
π π₯ β
π΄ π₯ + β β π΄ π₯
β
In the limit, as the rectangle becomes infinitely thin, this becomes an
equality:
π π₯ = lim
ββ0
π΄ π₯ + β β π΄ π₯
β
But we know from differentiating by first principles that:
π΄β²
π₯ = lim
ββ0
π΄ π₯ + β β π΄ π₯
β
β΄ π π₯ = π΄β²(π₯)
And thus we have proven that the gradient of the area function is
indeed π(π₯) (and hence the integral of π(π₯) the area).
But weβre missing one final bit: Why does
β«
π
π
π π₯ ππ₯ give the area between π₯ = π and π₯ = π?
π π
The area between π and π is the area up to
π, i.e. π΄(π), with the area up to π, i.e. π΄ π ,
cut out. This gives π΄ π β π΄ π as expected.
Because we obtained π΄(π₯) by an integration,
we have a constant of integration +π. But
because of the subtraction π΄ π β π΄(π),
these constants in each of π΄(π) and π΄ π
cancel, which explains why the constant is
not present in definite integration.
This is known as the Fundamental Theorem of Calculus.
24. Exercise 13E
Pearson Pure Mathematics Year 1/AS
Pages 299-300
Extension
[MAT 2007 1H] Given a function π(π₯), you are
told that
0
1
3π π₯ ππ₯ +
1
2
2π π₯ ππ₯ = 7
0
2
π π₯ ππ₯ +
1
2
π π₯ ππ₯ = 1
It follows that β«
0
2
π(π₯) ππ₯ equals what?
Because the area between π = π and 2 is the
sum of the area between 0 and 1, and
between 1 and 2, it follows that
β«
π
π
π π π π = β«
π
π
π π π π + β«
π
π
π π π π. Also
note that β« ππ π π π = π β« π(π) π π
Letting π = β«
π
π
π π π π and π = β«
π
π
π π π π
purely for convenience, then:
ππ + ππ = π
π + π + π = π
Solve, π = π, π = βπ, β΄ π + π = π
[MAT 2011 1G]
A graph of the function π¦ = π π₯ is sketched
on the axes below:
What is the value of β«β1
1
π(π₯2
β 1) ππ₯?
We first need to reflect on what part of the
function π weβre actually using. π in the
integral varies between -1 and 1, thus ππ
β π,
i.e. the input of π, varies between -1 and 0.
Weβre therefore only using the left half of the
graph, and thus π π = π + π
β΄
βπ
π
π(ππ
β π) π π =
βπ
π
ππ
β π + π π π
=
π
π
ππ
βπ
π
=
π
π
β β
π
π
=
π
π
?
?
1
2
25. βNegative Areasβ
Sketch the curve π¦ = π₯(π₯ β 1)(π₯ β 2) (which expands to give π¦ = π₯3 β 3π₯2 + 2π₯).
Now calculate β«0
2
π₯ π₯ β 1 π₯ β 2 ππ₯. Why is this result surprising?
π
π
ππ
β πππ
+ ππ π π =
π
π
ππ
β ππ
+ ππ
π
π
= π
So the total βareaβ is 0!
π₯
π¦
1 2
ππ₯
π(π₯) Integration β« π π₯ ππ₯ is just the sum
of areas of infinitely thin rectangles,
where the current π¦ value (i.e. π(π₯)) is
each height, and the widths are ππ₯.
i.e. The area of each is π π₯ Γ ππ₯
The problem is, when π(π₯) is negative,
then π π₯ Γ ππ₯ is negative, i.e. a
negative area!
The result is that the βpositive areaβ
from 0 to 1 is cancelled out by the
βnegative areaβ from 1 to 2, giving an
overall βareaβ of 0.
So how do we resolve this?
Fro Note: This explains the ππ₯ in
the β« π π₯ ππ₯, which effectively
means βthe sum of the areas of
strips, each of area π π₯ Γ ππ₯.
So the ππ₯ is not just part of the
β« notation, itβs behaving as a
physical quantity! (i.e. length)
?
26. Example
Find the total area bound between the curve π¦ = π₯(π₯ β 1)(π₯ β 2) and the π₯-axis.
π₯
π¦
1 2
Strategy:
Separately find the area between π₯ = 0
and 1, and between 1 and 2. Treat any
negative areas as positive.
π π β π π β π = ππ
β πππ
+ ππ
π
π
ππ β πππ + ππ π π =
π
π
ππ β ππ + ππ
π
π
=
π
π
π
π
ππ β πππ + ππ π π =
π
π
ππ β ππ + ππ
π
π
= β
π
π
Treating both as positive:
π¨πππ =
π
π
+
π
π
=
π
π
?
? Solution
28. πΌ π
Exercise 13F
Pearson Pure Mathematics Year 1/AS
Pages 301-302
Extension
[MAT 2010 1I] For a positive number π, let
πΌ π =
0
π
4 β 2π₯2
ππ₯
Then
ππΌ
ππ
= 0 when π is what value?
1
Hint: Itβs not actually even possible to
integrate 2π₯2
, but we can still sketch the
graph. Reflect on what
ππΌ
ππ
actually means.
π₯
π¦
π
π° π represents the area
from π = π up to π.
π¦ = 4 β 2π₯2
π π°
π π
represents the rate of change of area as π
increases. Thus if
π π°
π π
= π, the area is not
changing. This must happen at the π-intercept
of the graph, because once the curve goes
negative, the total area will start to decrease.
π β πππ
= π β π = π
The answer is π = π.
?
[STEP I 2014 Q3]
The numbers π and π, where π > π β₯ 0, are such that
π
π
π₯2
ππ₯ =
π
π
π₯ ππ₯
2
(i) In the case π = 0 and π > 0, find the value of π.
(ii) In the case π = 1, show that π satisfies
3π3
β π2
β 7π β 7 = 0
Show further, with the help of a sketch, that there
is only one (real) value of π that satisfies the
equation and that it lies between 2 and 3.
(iii) Show that 3π2
+ π2
= 3π2
π, where π = π + π
and π = π β π, and express π2
in terms of π.
Deduce that 1 < π β π β€
4
3
Guidance for this problem on next slide.
2
29. [STEP I 2014 Q3] The numbers π and π, where π > π β₯ 0, are such that β«
π
π
π₯2
ππ₯ = β«
π
π
π₯ ππ₯
2
(i) In the case π = 0 and π > 0, find the value of π.
(ii) In the case π = 1, show that π satisfies
3π3
β π2
β 7π β 7 = 0
Show further, with the help of a sketch, that there is only one (real) value of π that satisfies
the equation and that it lies between 2 and 3.
(iii) Show that 3π2
+ π2
= 3π2
π, where π = π + π and π = π β π, and express π2
in terms of π.
Deduce that 1 < π β π β€
4
3
Guidance for Extension Problem 2
30. Areas between curves and lines
π¦ = π₯
π¦
?
How could we find the area
between the line and the curve?
Start with the area under
π = π π β π up to the
point of intersection, then
subtract the area of the
triangle to βcut it outβ.
Click for
Fro-animation >
?
Determine the area between the lines with equations π¦ = π₯ 4 β π₯ and π¦ = π₯
Find point of intersection:
π₯ 4 β π₯ = π₯
β΄ π₯ = 0 ππ π₯ = 3
Area under curve:
0
3
π₯ 4 β π₯ ππ₯ = 2π₯2
β
1
3
π₯3
0
3
= 9
Area of triangle =
1
2
Γ 3 Γ 3 =
9
2
β΄ Shaded area = 9 β
9
2
=
9
2
?
32. Test Your Understanding
Edexcel C2 May 2012 Q5
π΄ 2,8 , π΅ 9,1
57
1
6
or
343
6
? a
? b
Alternative Method:
If the top curve has equation π¦ = π π₯
and the bottom curve π¦ = π π₯ , the
area between them is:
π
π
π π₯ β π π₯ ππ₯
This means you can integrate a single
expression to get the final area,
without any adjustment required after.
a
b
33. Exercise 13G
Pearson Pure Mathematics Year 1/AS
Pages 304-306
Extension
[MAT 2005 1A] What is the area of the region
bounded by the curves π¦ = π₯2
and π¦ = π₯ + 2?
π
π
[MAT 2016 1H] Consider two functions
π π₯ = π β π₯2
π π₯ = π₯4
β π
For precisely which values of π > 0 is the area of
the region bounded by the π₯-axis and the curve
π¦ = π π₯ bigger than the area of the region
bounded by the π₯-axis and the curve π¦ = π π₯ ?
(Your answer should be an inequality in terms of π)
1
2
(Official solution)
The area bounded by the π-axis and
the curve π = π π , π¨π is equal to
π¨π =
π
π
π(π) π π =
π
π
π
π
π
whilst the area bounded by the π-axis
and the curve π = π π , π¨π is equal
to
π¨π =
βπ
π
π
π
π π π π =
π
π
π
π
π
We require an π such that π¨π > π¨π so
π
π
π
π
π >
π
π
π
π
π
πππ
π
π > πππ
π
π
π
π
π >
π
π
and so the answer is (e).
? 2
? 1