Slides: C2 - Chapter 11 - Integration

advertisement
C2 Chapter 11 Integration
Dr J Frost (jfrost@tiffin.kingston.sch.uk)
Last modified: 17th October 2013
Recap
2ð‘Ĩ 2
2 3 3 2
+ 3ð‘Ĩ 𝑑ð‘Ĩ = ð‘Ĩ + ð‘Ĩ + ?
𝑐
3
2
1+ð‘Ĩ+
ð‘Ĩ
7
−3
ð‘Ĩ2
+
ð‘Ĩ3
1 2 1 3 1 4
𝑑ð‘Ĩ = ð‘Ĩ + ð‘Ĩ + ð‘Ĩ ? + ð‘Ĩ + 𝑐
2
3
4
3 −4
𝑑ð‘Ĩ = − ð‘Ĩ 3 + 𝑐
4
?
1
2+ ð‘Ĩ
−2
−1
𝑑ð‘Ĩ = −2ð‘Ĩ − 2ð‘Ĩ ?+ 𝑐
2
ð‘Ĩ
ð‘Ĩ ð‘Ĩ
2 −3
𝑑ð‘Ĩ = − ð‘Ĩ 2 + 𝑐
4
ð‘Ĩ
3
?
Definite Integration
ð‘Ķ
Suppose you wanted to find
the area under the curve
between ð‘Ĩ = 𝑎 and ð‘Ĩ = 𝑏.
𝑎
ð›ŋð‘Ĩ
𝑏
ð‘Ĩ
We could add together the area of individual strips,
which we want to make as thin as possible…
Definite Integration
ð‘Ķ
ð‘Ķ = 𝑓(ð‘Ĩ)
ð›ŋð‘Ĩ
ð‘Ĩ1
𝑎
ð‘Ĩ2
ð‘Ĩ3 ð‘Ĩ4 ð‘Ĩ5 ð‘Ĩ6
ð‘Ĩ
ð‘Ĩ7
𝑏
What is the total area between ð‘Ĩ1 and ð‘Ĩ7 ?
7
𝑏
𝑓 ð‘Ĩ𝑖 ð›ŋð‘Ĩ
𝑖=1
𝑓 ð‘Ĩ 𝑑ð‘Ĩ
As ð›ŋð‘Ĩ → 0
𝑎
Definite Integration
𝑏
𝑓 ð‘Ĩ 𝑑ð‘Ĩ
𝑎
You could think of this as “Sum the values of 𝑓(ð‘Ĩ) between ð‘Ĩ = 𝑎 and ð‘Ĩ = 𝑏.”
ð‘Ķ
Reflecting on above, do you think the
following definite integrals would be
positive or negative or 0?
ð‘Ķ = sin ð‘Ĩ
𝜋
2
−
ïƒŧ
+

0ïƒŧ
sin ð‘Ĩ 𝑑ð‘Ĩ
−
ïƒŧ
+ïƒŧ
0

sin ð‘Ĩ 𝑑ð‘Ĩ
−

+ïƒŧ
0ïƒŧ
sin ð‘Ĩ 𝑑ð‘Ĩ
0
𝜋
2𝜋
ð‘Ĩ
2𝜋
0
2𝜋
𝜋
2
Evaluating Definite Integrals
2
3ð‘Ĩ 2 𝑑ð‘Ĩ
1
= ð‘Ĩ 3? 12
= 23 −? 13
=7
?
𝑏
𝑎
𝑓 ′ ð‘Ĩ 𝑑ð‘Ĩ = 𝑓 ð‘Ĩ
𝑏
𝑎
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.
= 𝑓 𝑏 − 𝑓(𝑎)
Evaluating Definite Integrals
2
2ð‘Ĩ 3 + 2ð‘Ĩ 𝑑ð‘Ĩ
1
1 4
= ð‘Ĩ + ð‘Ĩ2
2
2
1
1
= 8 + 4 −? + 1
2
21
=
2
−1
4ð‘Ĩ 3 + 3ð‘Ĩ 2 𝑑ð‘Ĩ
−2
= ð‘Ĩ 4 + ð‘Ĩ 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 ð‘Ķ = 𝑓 ð‘Ĩ the ð‘Ĩ-axis and the lines
ð‘Ĩ = 𝑎 and ð‘Ĩ = 𝑏.
a 𝑓 ð‘Ĩ = 3ð‘Ĩ 2 − 2ð‘Ĩ + 2
c 𝑓 ð‘Ĩ = ð‘Ĩ + 2ð‘Ĩ
8
e 𝑓 ð‘Ĩ = ð‘Ĩ3 + ð‘Ĩ
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 = ð‘Ĩ(ð‘Ĩ 2 − 4). Find the area of the
shaded region (hint: first find the roots).
𝟒
?
4
6
Find the area of the finite region between the curve with equation
ð‘Ķ = (3 − ð‘Ĩ)(1 + ð‘Ĩ) and the ð‘Ĩ-axis.
𝟐
𝟏𝟎?
𝟑
Find the area of the finite region between the curve with equation ð‘Ķ = ð‘Ĩ 2 2 − ð‘Ĩ
and the ð‘Ĩ-axis.
𝟏
𝟏?
𝟑
Harder Examples
Find the area bounded between the curve with equation ð‘Ķ = ð‘Ĩ 3 −
ð‘Ĩ and the ð‘Ĩ-axis.
ð‘Ķ
Sketch:
(Hint: factorise!)
?
−1
1
ð‘Ĩ
1
Looking at the sketch, what is −1 ð‘Ĩ 3 − ð‘Ĩ 𝑑ð‘Ĩ 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 𝑑ð‘Ĩ = − 1
ð‘Ĩ
ð‘Ĩ
−
3
𝑑ð‘Ĩ
=
+
−1
0
𝟏
𝟒
So total area is +
4
𝟏
𝟏
=
𝟒
𝟐
?
4
Harder Examples
Sketch the curve with equation ð‘Ķ = ð‘Ĩ ð‘Ĩ − 1 ð‘Ĩ + 3 and find the area
between the curve and the ð‘Ĩ-axis.
The Sketch
The number crunching
ð‘Ĩ ð‘Ĩ − 1 ð‘Ĩ + 3 = ð‘Ĩ 3 − 2ð‘Ĩ 2 − 3ð‘Ĩ
ð‘Ķ
0
−3
1
?
-3
1
ð‘Ĩ
0
ð‘Ĩ 3 − 2ð‘Ĩ 2 − 3ð‘Ĩ 𝑑ð‘Ĩ = 11.25
ð‘Ĩ3
−
2ð‘Ĩ 2
7
− 3ð‘Ĩ 𝑑ð‘Ĩ = −
?
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 ð‘Ĩ-axis.
1
ð‘Ķ =ð‘Ĩ ð‘Ĩ+2
2
ð‘Ķ = ð‘Ĩ+1 ð‘Ĩ−4
3
ð‘Ķ = ð‘Ĩ+3 ð‘Ĩ ð‘Ĩ−3
ð‘Ĩ2
4
ð‘Ķ=
ð‘Ĩ−2
5
ð‘Ķ =ð‘Ĩ ð‘Ĩ−2 ð‘Ĩ−5
1
1?
3
5
20
?6
1
40?
2
1
1?
3
1
21?
12
Curves bound between two lines
ð‘Ķ = 𝑓(ð‘Ĩ)
ð›ŋð‘Ĩ
ð‘Ĩ
𝑎
𝒃
𝑏
Remember that 𝒂 𝒇(𝒙) meant the sum of all the ð‘Ķ values
between ð‘Ĩ = 𝑎 and ð‘Ĩ = 𝑏 (by using infinitely thin strips).
Curves bound between two lines
𝑏
𝑎
ð‘Ĩ
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 ð‘Ĩ −
? 𝑓(ð‘Ĩ)
𝑏
therefore
area…
ðī=
𝑎
𝑔 ð‘Ĩ ?− 𝑓 ð‘Ĩ
Curves bound between two lines
Find the area bound between
ð‘Ķ = ð‘Ĩ and ð‘Ķ = ð‘Ĩ 4 − ð‘Ĩ .
ð‘Ķ
3
ð‘Ĩ 4−ð‘Ĩ ?
− ð‘Ĩ 𝑑ð‘Ĩ = 4.5
0
ð‘Ĩ
Bro Tip: Always do the function of the top
line minus the function of the bottom line.
That way the difference in the ð‘Ķ 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)
ð‘Ĩ = −4,
ð‘Ĩ =?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
ð‘Ļ𝒓𝒆𝒂 = 𝟏𝟔?
𝟑
Exercise 11D
1 A region is bounded by the line ð‘Ķ = 6 and the curve ð‘Ķ = ð‘Ĩ 2 + 2.
a) Find the coordinates of the points of intersection.
b) Hence find the area of the finite region bounded by ðīðĩ and the curve.
3 The diagram shows a sketch of part of the curve with equation
2
3
ð‘Ķ = 9 − 3ð‘Ĩ − 5ð‘Ĩ − ð‘Ĩ and the line with equation ð‘Ķ = 4 − 4ð‘Ĩ.
The line cuts the curve at the points ðī −1,8 and ðĩ 1,0 .
Find the area of the shaded region between ðīðĩ and the curve.
ðī −2,6 ðĩ 2,6
2
ðī𝑟𝑒𝑎 = 10
3
?
ðī
ðĩ
2
3
?
6
4
Find the area of the finite region bounded by the curve with equation ð‘Ķ =
1 − ð‘Ĩ ð‘Ĩ + 3 and the line ð‘Ķ = ð‘Ĩ + 3.
4.5
?
9 The diagram shows part of the curve with equation ð‘Ķ = 3 ð‘Ĩ − ð‘Ĩ 3 + 4 and the line
1
with equation ð‘Ķ = 4 − 2 ð‘Ĩ.
a) Verify that the line and the curve cross at ðī 4,2 .
b) Find the area of the finite region bounded by the curve and the line. 4
?
7.2
ðī
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 ð‘Ķ =
ð‘Ĩ 2 + 1 and the line with equation ð‘Ķ = 7 − ð‘Ĩ.
a) Find the area of 𝑅1 .
b) Find the area of 𝑅2 .
ð‘Ķ
7
𝑅1
5
6
1
𝑅2 = 17
6
𝑅1 = 20
?
𝑅2
7
ð‘Ĩ
Trapezium Rule
y4
y3
What is the area here?
y2
ðī𝑟𝑒𝑎
1
= ℎ ð‘Ķ1 + ð‘Ķ2
2
1 ?
+ ℎ ð‘Ķ2 + ð‘Ķ3
2
1
+ ℎ ð‘Ķ3 + ð‘Ķ4
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
𝑏
𝑎
ℎ
ð‘Ķ 𝑑ð‘Ĩ ≈ ð‘Ķ1 + 2 ð‘Ķ2 + â‹Ŋ + ð‘Ķ𝑛−1 + ð‘Ķ𝑛
2
Area under curve
is approximately
Example
We’re approximating the region bounded between ð‘Ĩ = 1,
ð‘Ĩ = 3, the x-axis the curve ð‘Ķ = ð‘Ĩ 2
x
1
1.5
2
2.5
3
y
1
2.25
4
6.25
9
?
ℎ = 0.5
?
ðī𝑟𝑒𝑎 ≈ 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
?
ð‘Ļ𝒓𝒆𝒂 =
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?
Download