1. Dr J Frost (jfrost@tiffin.kingston.sch.uk)
C2 Chapter 11 Integration
Dr J Frost
Last modified: 17th October 2013

2. Recap
2 𝑥 2 +3𝑥 𝑑𝑥 = 2 3 𝑥 𝑥 2 +𝑐
1+𝑥+ 𝑥 2 + 𝑥 3 𝑑𝑥 =𝑥 𝑥 𝑥 𝑥 4 +𝑐
𝑥 − 𝑑𝑥 =− 3 4 𝑥 − 𝑐
2+ 𝑥 𝑥 2 𝑑𝑥 =−2 𝑥 −1 −2 𝑥 − 𝑐
𝑥 𝑥 𝑥 4 𝑑𝑥 =− 2 3 𝑥 − 𝑐 ? ? ? ? ?

3. Definite Integration 𝑦
Suppose you wanted to find the area under the curve between 𝑥=𝑎 and 𝑥=𝑏. 𝛿𝑥 𝑥 𝑎 𝑏
We could add together the area of individual strips, which we want to make as thin as possible…

4. Definite Integration What is the total area between 𝑥 1 and 𝑥 7 ?𝑦
𝑦=𝑓(𝑥) 𝛿𝑥 𝑥
𝑥 1
𝑥 2
𝑥 3
𝑥 4
𝑥 5
𝑥 6
𝑥 7 𝑎 𝑏
What is the total area between 𝑥 1 and 𝑥 7 ?
𝑖=1 7 𝑓 𝑥 𝑖 𝛿𝑥
𝑎 𝑏 𝑓 𝑥 𝑑𝑥
As 𝛿𝑥→0

5. Definite Integration 𝑎 𝑏 𝑓 𝑥 𝑑𝑥          − + − + − +
𝑎 𝑏 𝑓 𝑥 𝑑𝑥
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
0 𝜋 2 sin 𝑥 𝑑𝑥  − +   𝑥 𝜋 2𝜋
0 2𝜋 sin 𝑥 𝑑𝑥  − +  
𝜋 2 2𝜋 sin 𝑥 𝑑𝑥 −  +  

6. Evaluating Definite Integrals
1 2 3 𝑥 2 𝑑𝑥 ?
= 𝑥
We use square brackets to say that we’ve integrated the function, but we’re yet to involve the limits 1 and 2.
= 2 3 − 1 3 =7 ?
Then we find the difference when we sub in our limits. ?
𝑎 𝑏 𝑓 ′ 𝑥 𝑑𝑥= 𝑓 𝑥 𝑎 𝑏 =𝑓 𝑏 −𝑓(𝑎)

7. Evaluating Definite Integrals
1 2 2 𝑥 3 +2𝑥 𝑑𝑥 = 𝑥 4 + 𝑥 = 8+4 − = 21 2
−2 −1 4𝑥 3 +3 𝑥 2 𝑑𝑥 = 𝑥 4 + 𝑥 3 −2 −1 = 1−1 − 16−8 =−8 ? ?
Bro Tip: Be careful with your negatives, and use bracketing to avoid errors.

8. 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.
𝑓 𝑥 =3 𝑥 2 −2𝑥 𝑎=0, 𝑏=2
𝑓 𝑥 = 𝑥 +2𝑥 𝑎=1, 𝑏=2
𝑓 𝑥 = 8 𝑥 𝑥 𝑎=1, 𝑏=4
The sketch shows the curve with equation y=𝑥( 𝑥 2 −4). Find the area of the shaded region (hint: first find the roots).
Find the area of the finite region between the curve with equation 𝑦=(3−𝑥)(1+𝑥) and the 𝑥-axis.
Find the area of the finite region between the curve with equation 𝑦= 𝑥 2 2−𝑥 and the 𝑥-axis. 𝟖
𝟒.𝟐𝟐
𝟏𝟎𝟏 𝟏𝟐 ? a ? c ? e 2 𝟒 ?
𝟏𝟎 𝟐 𝟑 ? 4 6
𝟏 𝟏 𝟑 ?

9. Harder Examples Sketch: ? ? ?
Find the area bounded between the curve with equation 𝑦= 𝑥 3 −𝑥 and the 𝑥-axis.
Sketch:
(Hint: factorise!) ? 𝑦 𝑥 −1 1
Looking at the sketch, what is −1 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.
−1 0 𝑥 3 −3 𝑑𝑥 =− 𝑥 3 −3 𝑑𝑥 =+ 1 4
So total area is 𝟏 𝟒 + 𝟏 𝟒 = 𝟏 𝟐 ?

10. Harder Examples ? ? The Sketch The number crunching
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 ? ? 𝑦
𝑥 𝑥−1 𝑥+3 = 𝑥 3 −2 𝑥 2 −3𝑥
−3 0 𝑥 3 −2 𝑥 2 −3𝑥 𝑑𝑥 =11.25
0 1 𝑥 3 −2 𝑥 2 −3𝑥 𝑑𝑥 =− 7 12
Adding: =11 5 6 𝑥 -3 1

11. Exercise 11C
Find the area of the finite region or regions bounded by the curves and the 𝑥-axis.
1 1 3 ?
𝑦=𝑥 𝑥+2
𝑦= 𝑥+1 𝑥−4
𝑦= 𝑥+3 𝑥 𝑥−3
𝑦= 𝑥 2 𝑥−2
𝑦=𝑥 𝑥−2 𝑥−5 1
20 5 6 ? 2
40 1 2 ? 3
1 1 3 ? 4 5 ?

12. Curves bound between two lines
𝑦=𝑓(𝑥) 𝛿𝑥 𝑥 𝑎 𝑏
Remember that 𝒂 𝒃 𝒇(𝒙) meant the sum of all the 𝑦 values between 𝑥=𝑎 and 𝑥=𝑏 (by using infinitely thin strips).

13. Curves bound between two lines
𝑦=𝑔(𝑥)
𝑦=𝑓(𝑥) 𝑥 𝑏 𝑎
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)
𝐴= 𝑎 𝑏 𝑔 𝑥 −𝑓 𝑥 𝑑𝑥 ? ?
therefore area…

14. Curves bound between two lines
Find the area bound between 𝑦=𝑥 and 𝑦=𝑥 4−𝑥 . 𝑦
𝑦=𝑥
0 3 𝑥 4−𝑥 −𝑥 𝑑𝑥=4.5 ?
𝑦=𝑥 4−𝑥 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.

15. Curves bound between two lines
Edexcel C2 May 2013 (Retracted) ?
𝑥=−4, 𝑥=2 ?
Area = 36

16. More complex areas C A B ? 𝑨𝒓𝒆𝒂=𝟏𝟔 𝟏 𝟑
Bro Tip: Sometimes we can subtract areas from others. e.g. Here we could start with the area of the triangle OBC.
y = x(x-3) C
y = 2x A B
𝑨𝒓𝒆𝒂=𝟏𝟔 𝟏 𝟑 ?

17. Exercise 11D ?
𝐴 −2,6 𝐵 2,6
𝐴𝑟𝑒𝑎=10 2 3 1
A region is bounded by the line 𝑦=6 and the curve 𝑦= 𝑥 2 +2.
Find the coordinates of the points of intersection.
Hence find the area of the finite region bounded by 𝐴𝐵 and the curve.
The diagram shows a sketch of part of the curve with equation
𝑦=9−3𝑥−5 𝑥 2 − 𝑥 3 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.
Find the area of the finite region bounded by the curve with equation 𝑦= 1−𝑥 𝑥+3 and the line 𝑦=𝑥+3.
The diagram shows part of the curve with equation 𝑦=3 𝑥 − 𝑥 and the line with equation 𝑦=4− 1 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. 3 𝐴 𝐵
6 2 3 ? 4 ?
4.5 9 4 𝐴 ?
7.2

18. Exercise 11D
(Probably more difficult than you’d see in an exam paper, but you never know…)
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 𝑅 1 .
b) Find the area of 𝑅 2 . Q6 𝑦 7 ?
𝑅 1 =20 5 6
𝑅 2 =17 1 6
𝑅 1
𝑅 2 𝑥 7

19. Trapezium Rule
Instead of infinitely thin rectangular strips, we might use trapeziums to approximate the area under the curve.
What is the area here? y1 y2 y3 y4 h ?
𝐴𝑟𝑒𝑎= 1 2 ℎ 𝑦 1 + 𝑦 ℎ 𝑦 2 + 𝑦 ℎ 𝑦 3 + 𝑦 4

20. Trapezium Rule ℎ=0.5 𝐴𝑟𝑒𝑎≈8.75 ? ? In general:
width of each trapezium
𝑎 𝑏 𝑦 𝑑𝑥≈ ℎ 2 𝑦 𝑦 2 +…+ 𝑦 𝑛−1 + 𝑦 𝑛
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
2.25 4
6.25 9
ℎ= 𝐴𝑟𝑒𝑎≈8.75 ? ?

21. Trapezium Rule ? ? May 2013 (Retracted) 0.8571
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 𝟐 𝟎.𝟕𝟎𝟕𝟏+𝟐 𝟎.𝟕𝟓𝟗𝟏+𝟎.𝟖𝟎𝟗𝟎+𝟎.𝟖𝟓𝟕𝟏+𝟎.𝟗𝟎𝟑𝟕 +𝟎.𝟗𝟒𝟖𝟕 =𝟎.𝟒𝟏𝟔 ?

22. To add: When do we underestimate and overestimate?