c2 chapter 11 integration

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C2 Chapter 11: Integration Dr J Frost ([email protected]) www.drfrostmaths.com Last modified: 1 st September 2015

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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 Presentation

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Page 1: C2 Chapter 11  Integration

C2 Chapter 11: Integration

Dr J Frost ([email protected])www.drfrostmaths.com

Last modified: 1st September 2015

Page 2: C2 Chapter 11  Integration

Recap

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Page 3: C2 Chapter 11  Integration

Definite Integration

๐‘Ž ๐‘๐‘ฅ

๐‘ฆSuppose you wanted to find the area under the curve between and .

๐›ฟ๐‘ฅ

We could add together the area of individual strips, which we want to make as thin as possibleโ€ฆ

Page 4: C2 Chapter 11  Integration

Definite Integration

๐‘ฅ1 ๐‘ฅ7 ๐‘ฅ

๐‘ฆ

๐‘ฅ2 ๐‘ฅ3 ๐‘ฅ4 ๐‘ฅ5 ๐‘ฅ6

๐›ฟ๐‘ฅ

๐‘ฆ= ๐‘“ (๐‘ฅ )

What is the total area between and ?

As โˆซ๐‘Ž

๐‘

๐‘“ (๐‘ฅ )๐‘‘๐‘ฅ

๐‘Ž ๐‘

Page 5: C2 Chapter 11  Integration

Definite Integration

โˆซ๐‘Ž

๐‘

๐‘“ (๐‘ฅ )๐‘‘๐‘ฅ

You could think of this as โ€œSum the values of between and .โ€

๐‘ฆ=sin ๐‘ฅ

๐‘ฅ

๐‘ฆ Reflecting on above, do you think the following definite integrals would be positive or negative or 0?

๐œ‹ 2๐œ‹

โˆซ0

๐œ‹2

sin (๐‘ฅ )๐‘‘๐‘ฅ +โˆ’

โˆซ0

2 ๐œ‹

sin (๐‘ฅ ) ๐‘‘๐‘ฅ

0

+โˆ’ 0

โˆซ๐œ‹2

2 ๐œ‹

sin (๐‘ฅ ) ๐‘‘๐‘ฅ +โˆ’ 0

Page 6: C2 Chapter 11  Integration

Evaluating Definite Integrals

โˆซ๐‘Ž

๐‘

๐‘“ โ€ฒ (๐‘ฅ )๐‘‘๐‘ฅ=[ ๐‘“ (๐‘ฅ ) ]๐‘Ž๐‘= ๐‘“ (๐‘)โˆ’ ๐‘“ (๐‘Ž)

โˆซ1

2

3๐‘ฅ2๐‘‘๐‘ฅยฟ [๐‘ฅ3 ]12

We use square brackets to say that weโ€™ve integrated the function, but weโ€™re yet to involve the limits 1 and 2.

Then we find the difference when we sub in our limits.

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Page 7: C2 Chapter 11  Integration

Evaluating Definite Integrals

โˆซ1

2

2๐‘ฅ3+2๐‘ฅ ๐‘‘๐‘ฅ โˆซโˆ’2

โˆ’1

4 ๐‘ฅ3+3 ๐‘ฅ2๐‘‘๐‘ฅ?

?

Bro Tip: Be careful with your negatives, and use bracketing to avoid errors.

Page 8: C2 Chapter 11  Integration

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Ÿ’

๐Ÿ๐ŸŽ ๐Ÿ๐Ÿ‘

๐Ÿ๐Ÿ๐Ÿ‘

???

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Page 9: C2 Chapter 11  Integration

Harder ExamplesFind the area bounded between the curve with equation and the -axis.

Sketch:(Hint: factorise!)

๐‘ฅ

๐‘ฆ

โˆ’1 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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Page 10: C2 Chapter 11  Integration

Harder ExamplesSketch the curve with equation and find the area between the curve and the -axis.

Adding:

-3 1 ๐‘ฅ

๐‘ฆ

The Sketch The number crunching

? ?

Page 11: C2 Chapter 11  Integration

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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Page 12: C2 Chapter 11  Integration

Curves bound between two lines

๐‘ฅ๐›ฟ๐‘ฅ

๐‘ฆ= ๐‘“ (๐‘ฅ )

๐‘Ž ๐‘

Remember that meant the sum of all the values between and (by using infinitely thin strips).

Page 13: C2 Chapter 11  Integration

Curves bound between two lines

๐‘ฅ

๐‘ฆ=๐‘” (๐‘ฅ)

๐‘Ž ๐‘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?

๐‘ฆ= ๐‘“ (๐‘ฅ )

? therefore areaโ€ฆ

๐ด=โˆซ๐‘Ž

๐‘

๐‘” (๐‘ฅ )โˆ’ ๐‘“ (๐‘ฅ )?

Page 14: C2 Chapter 11  Integration

Curves bound between two lines

๐‘ฅ

๐‘ฆ

๐‘ฆ=๐‘ฅ

๐‘ฆ=๐‘ฅ

(4โˆ’๐‘ฅ

)

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 donโ€™t have to worry about negative areas.

โˆซ0

3

๐‘ฅ (4โˆ’๐‘ฅ )โˆ’๐‘ฅ ๐‘‘๐‘ฅ=4.5

Bro Tip: Weโ€™ll need to find the points at which they intersect.

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Page 15: C2 Chapter 11  Integration

Curves bound between two linesEdexcel C2 May 2013 (Retracted)

๐‘ฅ=โˆ’4 ,

Area =

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Page 16: C2 Chapter 11  Integration

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.

๐‘จ๐’“๐’†๐’‚=๐Ÿ๐Ÿ” ๐Ÿ๐Ÿ‘?

Page 17: C2 Chapter 11  Integration

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

๐ด๐ต

4 ๐ด

6 23

4.5

7.2

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?

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Page 18: C2 Chapter 11  Integration

Exercise 11D(Probably more difficult than youโ€™d see in an exam paper, but you never knowโ€ฆ)

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

๐‘ฅ

๐‘ฆ

7

7

๐‘…1

๐‘…2

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Page 19: C2 Chapter 11  Integration

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?

๐ด๐‘Ÿ๐‘’๐‘Ž=12 h ( ๐‘ฆ1+๐‘ฆ2 )+12 h ( ๐‘ฆ2+๐‘ฆ3 )+ 12 h ( ๐‘ฆ3+ ๐‘ฆ4 )?

Page 20: C2 Chapter 11  Integration

Trapezium RuleIn general:

โˆซ๐‘Ž

๐‘

๐‘ฆ ๐‘‘๐‘ฅ โ‰ˆ h2 (๐‘ฆ1+2 (๐‘ฆ 2+โ€ฆ+ ๐‘ฆ๐‘›โˆ’1 )+ ๐‘ฆ๐‘› )

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

Weโ€™re approximating the region bounded between , , the x-axis the curve

h=0.5 ๐ด๐‘Ÿ๐‘’๐‘Žโ‰ˆ 8.75?

Example

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Page 21: C2 Chapter 11  Integration

Bro Tip: You can generate table with Casio calcs . . Use โ€˜Alphaโ€™ button to key in X within the function. Press =

0.8571?

๐‘จ๐’“๐’†๐’‚=๐ŸŽ .๐Ÿ๐Ÿ (๐ŸŽ .๐Ÿ•๐ŸŽ๐Ÿ•๐Ÿ+๐Ÿ (๐ŸŽ .๐Ÿ•๐Ÿ“๐Ÿ—๐Ÿ+๐ŸŽ .๐Ÿ–๐ŸŽ๐Ÿ—๐ŸŽ+๐ŸŽ .๐Ÿ–๐Ÿ“๐Ÿ•๐Ÿ+๐ŸŽ .๐Ÿ—๐ŸŽ๐Ÿ‘๐Ÿ• )+๐ŸŽ .๐Ÿ—๐Ÿ’๐Ÿ–๐Ÿ• )=๐ŸŽ .๐Ÿ’๐Ÿ๐Ÿ”?

Trapezium RuleMay 2013 (Retracted)

Page 22: C2 Chapter 11  Integration

To add: When do we underestimate and overestimate?