P1 Chapter 13 :: Integration
Transcript of P1 Chapter 13 :: Integration
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Chapter Overview
1:: Find ๐ฆ given ๐๐ฆ
๐๐ฅ 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 ๐ฆ = ๐ฅ2 โ 2๐ฅ and the ๐ฅ-axis.
A curve has the gradient function
๐๐ฆ
๐๐ฅ= 3๐ฅ + 1
If the curve goes through the point 2,3 , determine ๐ฆ.
3:: Find areas bound between two different lines.
Find the points of intersection of ๐ฆ = ๐ฅ2 โ 4๐ฅ + 3 and ๐ฆ = ๐ฅ + 9, and hence find the area bound between the two lines.
Integrating ๐ฅ๐ terms
Integration is the opposite of differentiation.(For this reason it is also called โantidifferentiationโ)
5๐ฅ3multiply by power reduce power by 1
15๐ฅ2
increase power by 1divide by new power
Find ๐ฆ when ๐๐ฆ
๐๐ฅ= 3๐ฅ2
Adding 1 to the power and dividing by this power give us:๐ฆ = ๐ฅ3
However, other functions would also have differentiated to 3๐ฅ2:๐ฆ = ๐ฅ3 + 1, ๐ฆ = ๐ฅ3 โ 4, โฆ
Clearly we could have had any constant, as it disappears upon differentiation.
โด ๐ = ๐๐ + ๐
However, thereโs one added complicationโฆ
FunctionGradient Function
๐ is known as a constant of integration
Textbook Error on Pg289: โWhen integrating polynomialsโ, but Example 3 is not a polynomial because it has negative and fractional powers.
Examples
Find ๐ฆ when:
๐๐ฆ
๐๐ฅ= 4๐ฅ3 ๐ฆ = ๐ฅ4 + ๐
๐๐ฆ
๐๐ฅ= ๐ฅ5 ๐ฆ =
1
6๐ฅ6 + ๐
Exam Note: Historically โC1โ penalised the lack of +๐, but modules thereafter didnโt. You should always include it.
You could also write as ๐ฅ6
6. Itโs a
matter of personal preference.
๐๐ฆ
๐๐ฅ= 3๐ฅ
12 ๐ฆ = 3
2
3๐ฅ32 + ๐ = 2๐ฅ
32 + ๐
Fro Tip: Many students are taught to write 3๐ฅ
32
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.
More Fractional Examples
Find ๐ฆ when:
๐๐ฆ
๐๐ฅ=
4
๐ฅ= 4๐ฅโ
12 ๐ฆ = 8๐ฅ
12 + ๐
When we divide by 1
2we
multiply by the reciprocal, i.e. 2. Fro Tip: I recommend doing in your head when the simplification would be simple.
๐๐ฆ
๐๐ฅ= 5๐ฅโ2 ๐ฆ = โ5๐ฅโ1 + ๐
๐๐ฆ
๐๐ฅ= 4๐ฅ
23 ๐ฆ = 4
3
5๐ฅ53 + ๐ =
12
5๐ฅ53 + ๐
๐๐ฆ
๐๐ฅ= 10๐ฅโ
27 ๐ฆ = 10
7
5๐ฅ57 + ๐ = 14๐ฅ
57 + ๐
Test Your Understanding
Find ๐(๐ฅ) when:
๐โฒ ๐ฅ =2
๐ฅ7= 2๐ฅโ7 ๐ ๐ฅ = โ
1
3๐ฅโ6 + ๐
๐โฒ ๐ฅ = 3 ๐ฅ = ๐ฅ13 ๐ ๐ฅ =
3
4๐ฅ43 + ๐
๐โฒ ๐ฅ = 33๐ฅ56 ๐ ๐ฅ = 18๐ฅ
116 + ๐
๐โฒ ๐ฅ = 2๐ฅ + 7 ๐ ๐ฅ = ๐ฅ2 + 7๐ฅ + ๐
๐โฒ ๐ฅ = ๐ฅ2 โ 1 ๐ ๐ฅ =1
3๐ฅ3 โ ๐ฅ + ๐
Note: In case youโre wondering what happens if ๐๐ฆ
๐๐ฅ=
1
๐ฅ= ๐ฅโ1 , the problem is that after
adding 1 to the power, weโd be dividing by 0. You will learn how to integrate 1
๐ฅin Year 2.
Integration notation
The following notation could be used to differentiate an expression:
๐
๐๐ฅ5๐ฅ2 = 10๐ฅ
There is similarly notation for integrating an expression:
โซ 10๐ฅ ๐๐ฅ = 5๐ฅ2 + ๐
The ๐๐ฅ here means differentiating โwith respect to ๐ฅโ.
โIntegrateโฆโโโฆthis expressionโ
โโฆwith respect to ๐ฅโ(the ๐๐ฅ 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 weโll see later in the chapter.
It is called โindefiniteโ because the exact expression is unknown (due to the +๐).
Advanced Note: For the moment you should think of the just as
โซ
Examples
Find โซ(๐ฅโ3
2 + 2) ๐๐ฅ
Fro Note: The brackets are required if thereโs multiple terms.
= โ2๐ฅโ12 + 2๐ฅ + ๐
Find โซ(6๐ก2 โ 1) ๐๐ก
= 2๐ก3 โ ๐ก + ๐
Note the ๐๐ก instead of ๐๐ฅ.
Find โซ(๐๐ฅ3 + ๐) ๐๐ฅ where ๐ and ๐ are constants.
=1
4๐๐ฅ4 + ๐๐ฅ + ๐
Textbook (Minor) Error: โany other letters must be treated as constantsโ. Similar to the error in the differentiation chapter, it should read โany other letters, which are either constants or variables independent of ๐ฅ, can be treated as numbersโ. In โซ๐ฅ๐ฆ ๐๐ฅ,
if ๐ฆ is a variable, we can only treat ๐ฆ as a constant if it is not dependent on ๐ฅ, i.e. there is not some equation relating ๐ฆ to ๐ฅ.
Test Your Understanding
Edexcel C1 May 2014(R) Q4b
๐ฆ = 2๐ฅ5 + 6๐ฅโ12
เถฑ๐ฆ ๐๐ฅ =1
3๐ฅ6 + 12๐ฅ
12 + ๐
Finding constant of integration
Recall that when we integrate, we get a constant of integration, which could be any real value. This means we donโt know what the exact original function was.
๐ฅ
๐ฆ๐โฒ ๐ฅ = 3๐ฅ2
๐ฅ
๐ฆ
๐ ๐ฅ = ๐ฅ3
๐ ๐ฅ = ๐ฅ3 + ๐
๐ ๐ฅ = ๐ฅ3 โ 1
โซ
?1,3
But if we know one point on the curve, it leaves only one possibility.The curve with equation ๐ฆ = ๐(๐ฅ) passes
through 1,3 . Given that ๐โฒ ๐ฅ = 3๐ฅ2, find the equation of the curve.
๐โฒ ๐ฅ = 3๐ฅ2
โด ๐ ๐ฅ = ๐ฅ3 + ๐Using the point (1,3): 3 = 13 + ๐ โด ๐ = 2
๐ ๐ฅ = ๐ฅ3 + 2
Test Your Understanding
Edexcel C1 May 2014 Q10
To keep you occupied if you finish (a) quickly!
๐ฆ โ 25 = โ1
2๐ฅ โ 4(b)
Definite Integration
So far weโve seen integration as โthe opposite of differentiationโ, allowing us to find ๐ฆ = ๐ ๐ฅ when we know the gradient function ๐ฆ = ๐โฒ ๐ฅ .
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:
spee
d
time
๐ด๐๐๐ = ๐๐๐ ๐ก๐๐๐๐
เถฑ1
5
4๐ฅ3 ๐๐ฅ =
These are known as limits, which give the values of ๐ฅ weโre finding the area between.
๐ฆ
๐ฅ
๐ด๐๐๐ = ๐๐๐ ๐ก๐๐๐๐
1 5
๐ฆ = 4๐ฅ3
?
We integrate as normal, but put expression in square brackets, meaning we still need to evaluate the integrated expression using the limits.
Write โฆ โ (โฆ) and evaluate the expression for each of the limits, top one first.
๐ฅ4 15
= 54 โ 14
= 624
Another Example
เถฑโ3
3
๐ฅ2 + 1 ๐๐ฅ =1
3๐ฅ3 + ๐ฅ
โ3
3
=1
33 3 + 3 โ
1
3โ3 3 โ 3
= 12 โ โ12= 24
We DONโT have a constant of integration when doing definite integration. Iโll explain why later.
Write out you working EXACTLY as seen here. The โฆ โ (โฆ ) brackets are particularly crucial as youโll otherwise likely make a sign error.
โUse of Technologyโ Monkey says:
You can use the โซ๐๐โก button on your calculator to
evaluate definite integrals. But only use it to check your answer.
Problem Solving
Given that ๐ is a constant and โซ152๐๐ฅ + 7 ๐๐ฅ = 4๐2, show that there are two
possible values for ๐ and find these values.
เถฑ1
5
2๐๐ฅ + 7 ๐๐ฅ = ๐๐ฅ2 + 7๐ฅ 15
= 25๐ + 35 โ ๐ + 7= 24๐ + 28
โด 24๐ + 28 = 4๐2
๐2 โ 6๐ โ 7 = 0๐ + 1 ๐ โ 7 = 0๐ = โ1 ๐๐ 7
Remember: ๐ is a constant, so just treat it as a number.
Exercise 13D
Pearson Pure Mathematics Year 1/ASPage 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
๐ผ ๐ = เถฑ0
1
๐ฅ2 โ ๐ 2 ๐๐ฅ
as ๐ varies, is what?
๐ฐ ๐ = เถฑ๐
๐
๐๐ โ ๐๐๐๐ + ๐๐ ๐ ๐
=๐
๐๐๐ โ
๐
๐๐๐๐ +
๐
๐๐๐
๐
๐
=๐
๐โ๐
๐๐ + ๐๐
We then use differentiation or completing the square to find that the minimum value
of ๐
๐โ
๐
๐๐ + ๐๐ is
๐
๐๐.
[MAT 2015 1D] Let
๐ ๐ฅ = โซ01๐ฅ๐ก 2 ๐๐ก and ๐ ๐ฅ = โซ0
๐ฅ๐ก2 ๐๐ก
Let ๐ด > 0. Which of the following statements are true?
A) ๐ ๐ ๐ด is always bigger than ๐ ๐ ๐ด
B) ๐ ๐ ๐ด is always bigger than ๐ ๐ ๐ด
C) They are always equal.
D) ๐ ๐ ๐ด is bigger if ๐ด < 1, and ๐ ๐ ๐ด
is bigger if ๐ด > 1.
E) ๐ ๐ ๐ด is bigger if ๐ด < 1, and ๐ ๐ ๐ด
is bigger if ๐ด > 1.
(Official Sln) Evaluating: ๐ ๐ =๐๐
๐and ๐ ๐ =
๐๐
๐. Hence
๐ ๐ ๐จ =๐จ๐
๐๐and ๐ ๐ ๐จ =
๐จ๐
๐๐. Hence answer is (B).
1
2
Areas under curves
Earlier we saw that the definite integral โซ๐๐๐(๐) ๐ ๐ gives
the area between a positive curve ๐ฆ = ๐(๐ฅ), the ๐-axis, and the lines ๐ฅ = ๐ and ๐ฅ = ๐.(Weโll see why this works in a sec)
๐ฆ
๐ฅ
๐ด๐๐๐ = ๐๐๐ ๐ก๐๐๐๐
๐ ๐
๐ฆ = ๐(๐ฅ)
๐ด๐๐๐
Find the area of the finite region between the curve with equation ๐ฆ = 20 โ ๐ฅ โ ๐ฅ2 and the ๐ฅ-axis.
๐ฆ
๐ฅ
Factorise in order to find roots:
๐๐ โ ๐ โ ๐๐ = ๐๐๐ + ๐ โ ๐๐ = ๐๐ + ๐ ๐ โ ๐ = ๐๐ = โ๐ ๐๐ ๐ = ๐
Therefore area between curve and ๐-axis is:
เถฑ๐
๐
๐๐ โ ๐ โ ๐๐ ๐ ๐ = ๐๐๐ โ๐
๐๐๐ โ
๐
๐๐๐
โ๐
๐
= ๐๐ โ ๐ โ๐๐
๐โ โ๐๐๐ โ
๐๐
๐+๐๐๐
๐
=๐๐๐
๐
โ5 4
Just for your interestโฆ
Why does integrating a function give you the area under the graph?
Part 1:Youโre 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 youโre adding on each time.
Hereโs some practical examples using formulae you covered in your younger years!
๐๐ If you wanted to consider the rate at which the area of a circle increases with respect to the radius, consider a small change ๐๐ in the radius.The change in area is an infinitely thin ring, which looks like youโve drawn a circumference.So what is this rate?
๐ด = ๐๐2
โด๐๐ด
๐๐= 2๐๐
OMG THAT IS THE CIRCUMFERENCE
๐
So to draw a shaded circle (which has area!), itโs 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 canโt add a length to an area (only an area to an area, innit).But by considering the infinitely thin width ๐๐ of the circumference youโre drawing, it does have area!
๐๐ด
๐๐= 2๐๐ โ ๐๐ด = 2๐๐ ๐๐
i.e. the change in area, ๐๐ด, is 2๐๐ ร ๐๐If we โroll outโ the added area weโre adding each time, this forms a rectangle, whose length hopefully corresponds to the circumference of the circle:
?
๐๐
If the added area is 2๐๐ ๐๐ and the thickness is ๐๐, then the
length is 2๐๐ ๐๐
๐๐= 2๐๐ as expected.
Since ๐ ๐ฅ = ๐ดโฒ(๐ฅ), the area function ๐ด(๐ฅ) is the integral of ๐(๐ฅ). Thus:
เถฑ๐
๐
๐ ๐ฅ ๐๐ฅ = ๐ด ๐ฅ ๐๐ = ๐ด ๐ โ ๐ด(๐)
Part 2:It gets better!
Consider the volume of a sphere:
๐ =4
3๐๐3
๐๐
๐๐= 4๐๐2
OMG THATโS THE SURFACE AREA
This works for a similar reason to before.
๐ฅ
๐(๐ฅ)
๐ด(๐ฅ)
And the same principle applies to area under a graph. If ๐ด(๐ฅ)gives the area up to ๐ฅ, then what weโre โadding on each timeโ (i.e. the gradient) is sort of like drawing a line of length ๐(๐ฅ) at the right-most end each time (or more accurately, an infinitely thin rectangle of area ๐ ๐ฅ ร โ). Thus if ๐(๐ฅ) is the gradient of the area, then conversely the area is the integral of ๐ ๐ฅ .
๐ฅ + โ
This gives us an intuitive sense of why it works, but we need a formal proof:
โ
Using the diagram above, the area up to ๐ฅ + โ, i.e. ๐ด(๐ฅ + โ), is approximately the area up to ๐ฅ plus the added rectangular strip:
๐ด ๐ฅ + โ โ ๐ด ๐ฅ + (๐ ๐ฅ ร โ)Rearranging:
๐ ๐ฅ โ๐ด ๐ฅ + โ โ ๐ด ๐ฅ
โIn the limit, as the rectangle becomes infinitely thin, this becomes an equality:
๐ ๐ฅ = limโโ0
๐ด ๐ฅ + โ โ ๐ด ๐ฅ
โBut we know from differentiating by first principles that:
๐ดโฒ ๐ฅ = limโโ0
๐ด ๐ฅ + โ โ ๐ด ๐ฅ
โโด ๐ ๐ฅ = ๐ดโฒ(๐ฅ)
And thus we have proven that the gradient of the area function is indeed ๐(๐ฅ) (and hence the integral of ๐(๐ฅ) the area).
But weโre missing one final bit: Why does
โซ๐๐๐ ๐ฅ ๐๐ฅ give the area between ๐ฅ = ๐ and ๐ฅ = ๐?
๐ ๐
The area between ๐ and ๐ is the area up to ๐, i.e. ๐ด(๐), with the area up to ๐, i.e. ๐ด ๐ , cut out. This gives ๐ด ๐ โ ๐ด ๐ as expected.Because we obtained ๐ด(๐ฅ) by an integration, we have a constant of integration +๐. But because of the subtraction ๐ด ๐ โ ๐ด(๐), these constants in each of ๐ด(๐) and ๐ด ๐cancel, which explains why the constant is not present in definite integration.
This is known as the Fundamental Theorem of Calculus.
Exercise 13E
Pearson Pure Mathematics Year 1/ASPages 299-300
Extension
[MAT 2007 1H] Given a function ๐(๐ฅ), you are told that
เถฑ0
1
3๐ ๐ฅ ๐๐ฅ + เถฑ1
2
2๐ ๐ฅ ๐๐ฅ = 7
เถฑ0
2
๐ ๐ฅ ๐๐ฅ + เถฑ1
2
๐ ๐ฅ ๐๐ฅ = 1
It follows that โซ02๐(๐ฅ) ๐๐ฅ equals what?
Because the area between ๐ = ๐ and 2 is the sum of the area between 0 and 1, and between 1 and 2, it follows that
โซ๐๐๐ ๐ ๐ ๐ = โซ๐
๐๐ ๐ ๐ ๐ + โซ๐
๐๐ ๐ ๐ ๐. Also
note that โซ๐๐ ๐ ๐ ๐ = ๐โซ๐(๐) ๐ ๐
Letting ๐ = โซ๐๐๐ ๐ ๐ ๐ and ๐ = โซ๐
๐๐ ๐ ๐ ๐
purely for convenience, then:๐๐ + ๐๐ = ๐๐ + ๐ + ๐ = ๐
Solve, ๐ = ๐, ๐ = โ๐, โด ๐ + ๐ = ๐
[MAT 2011 1G] A graph of the function ๐ฆ = ๐ ๐ฅ is sketched on the axes below:
What is the value of โซโ11๐(๐ฅ2 โ 1) ๐๐ฅ?
We first need to reflect on what part of the function ๐ weโre actually using. ๐ in the integral varies between -1 and 1, thus ๐๐ โ ๐, i.e. the input of ๐, varies between -1 and 0.Weโre therefore only using the left half of the graph, and thus ๐ ๐ = ๐ + ๐
โด เถฑโ๐
๐
๐(๐๐ โ ๐) ๐ ๐ = เถฑโ๐
๐
๐๐ โ ๐ + ๐ ๐ ๐
=๐
๐๐๐
โ๐
๐
=๐
๐โ โ
๐
๐=๐
๐
12
โNegative Areasโ
Sketch the curve ๐ฆ = ๐ฅ(๐ฅ โ 1)(๐ฅ โ 2) (which expands to give ๐ฆ = ๐ฅ3 โ 3๐ฅ2 + 2๐ฅ).
Now calculate โซ02๐ฅ ๐ฅ โ 1 ๐ฅ โ 2 ๐๐ฅ. Why is this result surprising?
เถฑ๐
๐
๐๐ โ ๐๐๐ + ๐๐ ๐ ๐ =๐
๐๐๐ โ ๐๐ + ๐๐
๐
๐
= ๐
So the total โareaโ is 0!
๐ฅ
๐ฆ
1 2๐๐ฅ
๐(๐ฅ) Integration โซ ๐ ๐ฅ ๐๐ฅ is just the sum of areas of infinitely thin rectangles, where the current ๐ฆ value (i.e. ๐(๐ฅ)) is each height, and the widths are ๐๐ฅ.i.e. The area of each is ๐ ๐ฅ ร ๐๐ฅ
The problem is, when ๐(๐ฅ) is negative, then ๐ ๐ฅ ร ๐๐ฅ is negative, i.e. a negative area!The result is that the โpositive areaโ from 0 to 1 is cancelled out by the โnegative areaโ from 1 to 2, giving an overall โareaโ of 0.So how do we resolve this?
Fro Note: This explains the ๐๐ฅ in the โซ๐ ๐ฅ ๐๐ฅ, which effectively
means โthe sum of the areas of strips, each of area ๐ ๐ฅ ร ๐๐ฅ. So the ๐๐ฅ is not just part of the โซ notation, itโs behaving as a physical quantity! (i.e. length)
Example
Find the total area bound between the curve ๐ฆ = ๐ฅ(๐ฅ โ 1)(๐ฅ โ 2) and the ๐ฅ-axis.
๐ฅ
๐ฆ
1 2
Strategy: Separately find the area between ๐ฅ = 0and 1, and between 1 and 2. Treat any negative areas as positive.
๐ ๐ โ ๐ ๐ โ ๐ = ๐๐ โ ๐๐๐ + ๐๐
เถฑ๐
๐
๐๐ โ ๐๐๐ + ๐๐ ๐ ๐ =๐
๐๐๐ โ ๐๐ + ๐๐
๐
๐
=๐
๐
เถฑ๐
๐
๐๐ โ ๐๐๐ + ๐๐ ๐ ๐ =๐
๐๐๐ โ ๐๐ + ๐๐
๐
๐
= โ๐
๐
Treating both as positive:
๐จ๐๐๐ =๐
๐+๐
๐=๐
๐
๐ผ ๐
Exercise 13F
Pearson Pure Mathematics Year 1/ASPages 301-302
Extension
[MAT 2010 1I] For a positive number ๐, let
๐ผ ๐ = เถฑ0
๐
4 โ 2๐ฅ2๐๐ฅ
Then ๐๐ผ
๐๐= 0 when ๐ is what value?
1
Hint: Itโs not actually even possible to
integrate 2๐ฅ2, but we can still sketch the
graph. Reflect on what ๐๐ผ
๐๐actually means.
๐ฅ
๐ฆ
๐
๐ฐ ๐ represents the area from ๐ = ๐ up to ๐.
๐ฆ = 4 โ 2๐ฅ2
๐ ๐ฐ
๐ ๐represents the rate of change of area as ๐
increases. Thus if ๐ ๐ฐ
๐ ๐= ๐, the area is not
changing. This must happen at the ๐-intercept of the graph, because once the curve goes negative, the total area will start to decrease.
๐ โ ๐๐๐= ๐ โ ๐ = ๐
The answer is ๐ = ๐.
[STEP I 2014 Q3]The numbers ๐ and ๐, where ๐ > ๐ โฅ 0, are such that
เถฑ๐
๐
๐ฅ2 ๐๐ฅ = เถฑ๐
๐
๐ฅ ๐๐ฅ
2
(i) In the case ๐ = 0 and ๐ > 0, find the value of ๐.(ii) In the case ๐ = 1, show that ๐ satisfies
3๐3 โ ๐2 โ 7๐ โ 7 = 0Show further, with the help of a sketch, that there is only one (real) value of ๐ that satisfies the equation and that it lies between 2 and 3.
(iii) Show that 3๐2 + ๐2 = 3๐2๐, where ๐ = ๐ + ๐and ๐ = ๐ โ ๐, and express ๐2 in terms of ๐.
Deduce that 1 < ๐ โ ๐ โค4
3
Guidance for this problem on next slide.
2
[STEP I 2014 Q3] The numbers ๐ and ๐, where ๐ > ๐ โฅ 0, are such that โซ๐๐๐ฅ2 ๐๐ฅ = โซ๐
๐๐ฅ ๐๐ฅ
2
(i) In the case ๐ = 0 and ๐ > 0, find the value of ๐.(ii) In the case ๐ = 1, show that ๐ satisfies
3๐3 โ ๐2 โ 7๐ โ 7 = 0Show further, with the help of a sketch, that there is only one (real) value of ๐ that satisfies the equation and that it lies between 2 and 3.
(iii) Show that 3๐2 + ๐2 = 3๐2๐, where ๐ = ๐ + ๐ and ๐ = ๐ โ ๐, and express ๐2 in terms of ๐.
Deduce that 1 < ๐ โ ๐ โค4
3
Guidance for Extension Problem 2
Areas between curves and lines
๐ฆ = ๐ฅ๐ฆ
?How could we find the area between the line and the curve?
Start with the area under ๐ = ๐ ๐ โ ๐ up to the point of intersection, then subtract the area of the triangle to โcut it outโ.
Click for Fro-animation >
Determine the area between the lines with equations ๐ฆ = ๐ฅ 4 โ ๐ฅ and ๐ฆ = ๐ฅ
Find point of intersection:๐ฅ 4 โ ๐ฅ = ๐ฅโด ๐ฅ = 0 ๐๐ ๐ฅ = 3
Area under curve:
เถฑ0
3
๐ฅ 4 โ ๐ฅ ๐๐ฅ = 2๐ฅ2 โ1
3๐ฅ3
0
3
= 9
Area of triangle =1
2ร 3 ร 3 =
9
2
โด Shaded area = 9 โ9
2=
9
2
A Harder One
๐
๐ถ
๐ด ๐ต
[Textbook] The diagram shows a sketch of the curve with equation ๐ฆ = ๐ฅ ๐ฅ โ 3 and the line with equation ๐ฆ = 2๐ฅ.Find the area of the shaded region ๐๐ด๐ถ.
What areas should we subtract this time?Start with triangle ๐ถ๐ฉ๐ช and subtract the area under the curve ๐จ๐ช.
First find points of intersection:๐ ๐ โ ๐ = ๐๐ โ ๐ = ๐ ๐๐ ๐ = ๐
When ๐ = ๐, ๐ = ๐๐ โ ๐ช ๐, ๐๐Also need to find the point ๐จ:
๐ ๐ โ ๐ = ๐ โ ๐จ(๐, ๐)
โด Area of triangle ๐ถ๐ฉ๐ช =๐
๐ร ๐ ร ๐๐ = ๐๐
Area under ๐จ๐ช: โซ๐๐๐ ๐ โ ๐ ๐ ๐ = โฆ =
๐๐
๐
โด Shaded area = ๐๐ โ๐๐
๐=
๐๐
๐
Test Your Understanding
Edexcel C2 May 2012 Q5
๐ด 2,8 , ๐ต 9,1
571
6or
343
6
Alternative Method:If the top curve has equation ๐ฆ = ๐ ๐ฅand the bottom curve ๐ฆ = ๐ ๐ฅ , the area between them is:
เถฑ๐
๐
๐ ๐ฅ โ ๐ ๐ฅ ๐๐ฅ
This means you can integrate a single expression to get the final area, without any adjustment required after.
a
b
Exercise 13G
Pearson Pure Mathematics Year 1/ASPages 304-306
Extension
[MAT 2005 1A] What is the area of the region bounded by the curves ๐ฆ = ๐ฅ2 and ๐ฆ = ๐ฅ + 2?
๐
๐
[MAT 2016 1H] Consider two functions๐ ๐ฅ = ๐ โ ๐ฅ2
๐ ๐ฅ = ๐ฅ4 โ ๐For precisely which values of ๐ > 0 is the area of the region bounded by the ๐ฅ-axis and the curve ๐ฆ = ๐ ๐ฅ bigger than the area of the region bounded by the ๐ฅ-axis and the curve ๐ฆ = ๐ ๐ฅ ?(Your answer should be an inequality in terms of ๐)
1
2
(Official solution)The area bounded by the ๐-axis and the curve ๐ = ๐ ๐ , ๐จ๐ is equal to
๐จ๐ = เถฑ๐
๐
๐(๐) ๐ ๐ =๐
๐๐๐๐
whilst the area bounded by the ๐-axis and the curve ๐ = ๐ ๐ , ๐จ๐ is equal to
๐จ๐ = เถฑโ๐ ๐
๐ ๐
๐ ๐ ๐ ๐ =๐
๐๐๐๐
We require an ๐ such that ๐จ๐ > ๐จ๐ so๐
๐๐๐๐ >
๐
๐๐๐๐
๐๐๐๐๐ > ๐๐๐
๐๐
๐๐๐ >
๐
๐and so the answer is (e).