c2 chapter 11 integration
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C2 Chapter 11 Integration. Dr J Frost ([email protected]) . Last modified: 17 th October 2013. Recap. ?. ?. ?. ?. ?. Definite Integration. Suppose you wanted to find the area under the curve between and . . - PowerPoint PPT PresentationTRANSCRIPT
C2 Chapter 11: Integration
Dr J Frost ([email protected])www.drfrostmaths.com
Last modified: 1st September 2015
Recap
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Definite Integration
๐ ๐๐ฅ
๐ฆSuppose you wanted to find the area under the curve between and .
๐ฟ๐ฅ
We could add together the area of individual strips, which we want to make as thin as possibleโฆ
Definite Integration
๐ฅ1 ๐ฅ7 ๐ฅ
๐ฆ
๐ฅ2 ๐ฅ3 ๐ฅ4 ๐ฅ5 ๐ฅ6
๐ฟ๐ฅ
๐ฆ= ๐ (๐ฅ )
What is the total area between and ?
As โซ๐
๐
๐ (๐ฅ )๐๐ฅ
๐ ๐
Definite Integration
โซ๐
๐
๐ (๐ฅ )๐๐ฅ
You could think of this as โSum the values of between and .โ
๐ฆ=sin ๐ฅ
๐ฅ
๐ฆ Reflecting on above, do you think the following definite integrals would be positive or negative or 0?
๐ 2๐
โซ0
๐2
sin (๐ฅ )๐๐ฅ +โ
โซ0
2 ๐
sin (๐ฅ ) ๐๐ฅ
0
+โ 0
โซ๐2
2 ๐
sin (๐ฅ ) ๐๐ฅ +โ 0
Evaluating Definite Integrals
โซ๐
๐
๐ โฒ (๐ฅ )๐๐ฅ=[ ๐ (๐ฅ ) ]๐๐= ๐ (๐)โ ๐ (๐)
โซ1
2
3๐ฅ2๐๐ฅยฟ [๐ฅ3 ]12
We use square brackets to say that weโve integrated the function, but weโre yet to involve the limits 1 and 2.
Then we find the difference when we sub in our limits.
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Evaluating Definite Integrals
โซ1
2
2๐ฅ3+2๐ฅ ๐๐ฅ โซโ2
โ1
4 ๐ฅ3+3 ๐ฅ2๐๐ฅ?
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Bro Tip: Be careful with your negatives, and use bracketing to avoid errors.
Exercise 11B
Find the area between the curve with equation the -axis and the lines and .
The sketch shows the curve with equation . Find the area of the shaded region (hint: first find the roots).
Find the area of the finite region between the curve with equation and the -axis.
Find the area of the finite region between the curve with equation and the -axis.
1
2
4
6
ace
๐
๐๐ ๐๐
๐๐๐
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Harder ExamplesFind the area bounded between the curve with equation and the -axis.
Sketch:(Hint: factorise!)
๐ฅ
๐ฆ
โ1 1?
Looking at the sketch, what is 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. So total area is
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Harder ExamplesSketch the curve with equation and find the area between the curve and the -axis.
Adding:
-3 1 ๐ฅ
๐ฆ
The Sketch The number crunching
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Exercise 11CFind the area of the finite region or regions bounded by the curves and the -axis.
1
2
3
4
5
1 13
20 56
40 12
1 13
21 112
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Curves bound between two lines
๐ฅ๐ฟ๐ฅ
๐ฆ= ๐ (๐ฅ )
๐ ๐
Remember that meant the sum of all the values between and (by using infinitely thin strips).
Curves bound between two lines
๐ฅ
๐ฆ=๐ (๐ฅ)
๐ ๐How 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?
๐ฆ= ๐ (๐ฅ )
? therefore areaโฆ
๐ด=โซ๐
๐
๐ (๐ฅ )โ ๐ (๐ฅ )?
Curves bound between two lines
๐ฅ
๐ฆ
๐ฆ=๐ฅ
๐ฆ=๐ฅ
(4โ๐ฅ
)
Find the area bound between and .
Bro Tip: Always do the function of the top line minus the function of the bottom line. That way the difference in the values is always positive, and you donโt have to worry about negative areas.
โซ0
3
๐ฅ (4โ๐ฅ )โ๐ฅ ๐๐ฅ=4.5
Bro Tip: Weโll need to find the points at which they intersect.
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Curves bound between two linesEdexcel C2 May 2013 (Retracted)
๐ฅ=โ4 ,
Area =
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y = x(
x-3)
y = 2x
A B
C
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.
๐จ๐๐๐=๐๐ ๐๐?
Exercise 11D
A region is bounded by the line and the curve .a) Find the coordinates of the points of intersection.b) Hence find the area of the finite region bounded by and the curve.
The diagram shows a sketch of part of the curve with equation and the line with equation . The line cuts the curve at the points and . Find the area of the shaded region between and the curve.
Find the area of the finite region bounded by the curve with equation and the line .
The diagram shows part of the curve with equation and the line with equation .a) Verify that the line and the curve cross at .b) Find the area of the finite region bounded by the curve and the line.
1
3
4
9
๐ด๐ต
4 ๐ด
6 23
4.5
7.2
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Exercise 11D(Probably more difficult than youโd see in an exam paper, but you never knowโฆ)
The diagram shows a sketch of part of the curve with equation and the line with equation .
a) Find the area of .b) Find the area of .
Q6
๐ฅ
๐ฆ
7
7
๐ 1
๐ 2
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y1
y2
y3
y4
h h h
Trapezium Rule
Instead of infinitely thin rectangular strips, we might use trapeziums to approximate the area under the curve.
What is the area here?
๐ด๐๐๐=12 h ( ๐ฆ1+๐ฆ2 )+12 h ( ๐ฆ2+๐ฆ3 )+ 12 h ( ๐ฆ3+ ๐ฆ4 )?
Trapezium RuleIn general:
โซ๐
๐
๐ฆ ๐๐ฅ โ h2 (๐ฆ1+2 (๐ฆ 2+โฆ+ ๐ฆ๐โ1 )+ ๐ฆ๐ )
width of each trapezium
Area under curve
is approximately
x 1 1.5 2 2.5 3
y 1 2.25 4 6.25 9
Weโre approximating the region bounded between , , the x-axis the curve
h=0.5 ๐ด๐๐๐โ 8.75?
Example
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Bro Tip: You can generate table with Casio calcs . . Use โAlphaโ button to key in X within the function. Press =
0.8571?
๐จ๐๐๐=๐ .๐๐ (๐ .๐๐๐๐+๐ (๐ .๐๐๐๐+๐ .๐๐๐๐+๐ .๐๐๐๐+๐ .๐๐๐๐ )+๐ .๐๐๐๐ )=๐ .๐๐๐?
Trapezium RuleMay 2013 (Retracted)
To add: When do we underestimate and overestimate?