0- Areas of Compound RegionsL5

8/12/2019 0- Areas of Compound RegionsL5

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Areas of compound Regions


2/39

Review

1. Find the points of intersection of

y=x2andy = 6x

2. Drawy=x2 0 x 2 , andy = 6x 2 x 6,on the same diagram.

3. On a different diagram draw both curves for

-4 x 4


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Review

2

2

1. 6

6 0

3 2 0

3,2

9,4

Points are -3,9 ,(2,4)

x x

x x

x x

x

y


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Composite Areas

1. Find the shaded area below, which

represents the region bounded byy=x2and

y = 6x and thexaxis.

1 2 3 4 5 6

1

2

3

4

1 2 3 4 5 6

1

2

3

4


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Method

1. Draw a picture

1 2 3 4 5 6

1

2

3

4

1 2 3 4 5 6

1

2

3

4


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Method

1. Draw a picture

2. Find any points of intersection

1 2 3 4 5 6

1

2

3

4

1 2 3 4 5 6

1

2

3

4


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Method

1. Draw a picture


3. Do the integral (If necessary).

1 2 3 4 5 6

1

2

3

4

1 2 3 4 5 6

1

2

3

4


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Method

1. Draw a picture


3. Do the integral (if necessary).

1 2 3 4 5 6

1

2

3

4

1 2 3 4 5 6

1

2

3

4


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Method

Find any points of intersection (2,4)

1 2 3 4 5 6

1

2

3

4

1 2 3 4 5 6

1

2

3

4

62 2

0 26A x dx x dx


11/39

Method


1 2 3 4 5 6

1

2

3

4

1 2 3 4 5 6

1

2

3

4

62 2

0 2

23

0

6

14 4

3 2

A x dx x dx

xA


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Method

1 2 3 4 5 6

1

2

3

4

1 2 3 4 5 6

1

2

3

4

62 2

0 2

23

0

6

14 4

3 2

8

0 83

A x dx x dx

xA

A


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Method


1 2 3 4 5 6

1

2

3

4

1 2 3 4 5 6

1

2

3

4

62 2

0 2

23

0

2

6

14 4

3 2

80 8

32

103

A x dx x dx

xA

A

x u


14/39

Method

Find any points of intersection (2,4) [review]

1 2 3 4 5 6

1

2

3

4

1 2 3 4 5 6

1

2

3

4

62 2

0 2

23

0

2

6

14 4

3 2

8

0 83

210

3

A x dx x dx

xA

A

x u


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Note if the required area is between two

curves then the easiest method is to do the

integral of the top curve

the integral ofthe bottom.

-4 -2 2 4

2.5

5

7.5

10

12.5

15

-4 -2 2 4

2.5

5

7.5

10

12.5

15

top


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-4 -2 2 4

2.5

5

7.5

10

12.5

15

-4 -2 2 4

2.5

5

7.5

10

12.5

15

Note if the required area is between two

curves then the easiest method is to do the

integral of the top curve

the integral ofthe bottom.

toptop


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Composite Areas

1. Find the shaded area below, which

represents the region bounded byy=x2and

y = 6x.

-4 -2 2 4

2.5

5

7.5

10

12.5

15

-4 -2 2 4

2.5

5

7.5

10

12.5

15


18/39

Method

1. Draw a picture


19/39

Method

1. Draw a picture

2. Find thexvalues of the points of

intersection


20/39

Method

1. Draw a picture


intersection

3. Do the integral (topbottom). Thex

values will be the limits


21/39

Method

1. Draw a picture


intersection

3. Do the integral (topbottom). Thex

values will be the limits


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Solution

2 2

36A x x dx

-4 -2 2 4

2.5

5

7.5

10

12.5

15

-4 -2 2 4

2.5

5

7.5

10

12.5

15


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Solution

2 23

22 3

3

6

62 3

A x x dx

x xA x

-4 -2 2 4

2.5

5

7.5

10

12.5

15

-4 -2 2 4

2.5

5

7.5

10

12.5

15


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Solution

2 23

22 3

3

6

62 3

8 9 2712 2 18

3 2 3

A x x dx

x xA x

A

-4 -2 2 4

2.5

5

7.5

10

12.5

15

-4 -2 2 4

2.5

5

7.5

10

12.5

15


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Solution

2 23

22 3

3

6

62 3

8 9 2712 2 18

3 2 3

1 17 13

3 2

A x x dx

x xA x

A

-4 -2 2 4

2.5

5

7.5

10

12.5

15

-4 -2 2 4

2.5

5

7.5

10

12.5

15


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Solution2 2

3

22 3

3

2

6

62 3

8 9 2712 2 18

3 2 3

1 17 13

3 25

206

A x x dx

x xA x

A

u

-4 -2 2 4

2.5

5

7.5

10

12.5

15

-4 -2 2 4

2.5

5

7.5

10

12.5

15


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Solution2 2

3

22 3

3

2

6

62 3

8 9 2712 2 18

3 2 3

1 17 13

3 25

206

A x x dx

x xA x

A

u

-4 -2 2 4

2.5

5

7.5

10

12.5

15

-4 -2 2 4

2.5

5

7.5

10

12.5

15


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NoteGreat care must be taken to deal with curves

which are partly above and below thex

axis

1. Find the area bounded byy=x2x6,

thex-axis, the linesx = 2 andx = 5

Draw this region


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axis


thex-axis, the linesx = 2 andx = 5

Draw this region


30/39



axis


thex-axis and the linesx = 2 andx = 5

-4 -2 2 4 6

-5

5

10

15

20

-4 -2 2 4 6

-5

5

10

15

20

3 5


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2 2

2 3

3 53 2 3 2

2 3

6 6

6 63 2 3 2

A x x dx x x dx

x x x xA x x

3 5


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2 2

2 3

3 53 2 3 2

2 3

51 1 12 3 6 2

6 6

6 63 2 3 2

9 8 125 25 99 18 2 12 30 9 18

2 3 3 2 213 11 13

A x x dx x x dx

x x x xA x x

A

3 5


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2 2

2 3

3 53 2 3 2

2 3

51 1 12 3 6 2

1 26 3

56

6 6

6 63 2 3 2

9 8 125 25 99 18 2 12 30 9 18

2 3 3 2 2

13 11 13

2 12

14

A x x dx x x dx

x x x xA x x

A

3 5


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2 2

2 3

3 53 2 3 2

2 3

51 1 12 3 6 2

1 26 3

56

6 6

6 63 2 3 2

9 8 125 25 99 18 2 12 30 9 18

2 3 3 2 213 11 13

2 12

14

A x x dx x x dx

x x x xA x x

A


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35

Sum& Differences of Areas

You try :Find area enclosed between

y=x2, y=(x4)2and the x-axis.

Solve simultaneouslyy = x2

y = (x

4)2

x2= (x 4)2x2= x2- 8x + 160 = - 8x + 16

8x = 16

x = 2

y

x0 4202

x2

dxArea = + (x

4)2

dx2

4

= x3

30

2 (x 4)33

+2

4

( )83

03= ( )

03

-83+

= 5 units213


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36

Sum & Differencesof Areas

You try :Find area enclosed between

y=x2and y=x+2.

Solve simultaneouslyy = x2

y = x + 2x2= x + 2x2- x 2 = 0

(x 2)(x + 1) = 0

x = -1x = 2

-1 2

-1

2

- x2dx

= 4 1/2 units2

-1

2

Area = (x + 2) dx

-12

= (x + 2 - x2) dx

x33

=-1

2x22

+ 2x -

( )= 2 + 4 8/3 ( )- 1/2- 2 + 1/3

y

x


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The area enclosed between

y= 25 x2and y= x+ 13

3 3

2

1 2

4 4

3

2

4

33 2

4

( ). (25 13).

(12 ).

12

3 2 57.2 to 1 dp

x x

x

x

y y dx x x dx

x x dx

x xx


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38

Find the area that exists between the curves

y= -x2+ 4x 4

y= x2- 8x + 12 in the domain 0 to 4Find the points of intersection

Quick sketch functions and areas

Determine the definite integrals (domain)

Evaluate the definite integrals to find the areas.

Area = 16 units 2


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Homework

Cambridge 12

Exercise 1F

Q1& Q2 abcd in bothQ3b

Q4a

Q6Q8 & 11 c&d in both

Documents

0- Areas of Compound RegionsL5