Areas(1) Area Below x axis

y = f(x)y

x

Areas(1) Area Below x axis

y

x

y = f(x)

a b

1A

Areas(1) Area Below x axis

y

x

y = f(x)

a b

1A

b

adxxf 0

Areas(1) Area Below x axis

y

x

y = f(x)

a b

1A

b

adxxf 0

b

adxxfA1

Areas(1) Area Below x axis

y

x

y = f(x)

a b

1A

b

adxxf 0

b

adxxfA1

c2A

Areas(1) Area Below x axis

y

x

y = f(x)

a b

1A

b

adxxf 0

b

adxxfA1

c2A

c

bdxxf 0

Areas(1) Area Below x axis

y

x

y = f(x)

a b

1A

b

adxxf 0

b

adxxfA1

c2A

c

bdxxf 0

c

bdxxfA2

Area below x axis is given by;

Area below x axis is given by;

c

bdxxfA

Area below x axis is given by;

c

bdxxfA

OR

c

bdxxf

Area below x axis is given by;

c

bdxxfA

OR

c

bdxxf

OR

b

cdxxf

e.g. (i) y

x

3xy

1-1

e.g. (i) y

x

3xy

1-1

1

0

30

1

3 dxxdxxA

e.g. (i) y

x

3xy

1-1

1

0

30

1

3 dxxdxxA

10401

4

41

41 xx

e.g. (i) y

x

3xy

1-1

1

0

30

1

3 dxxdxxA

10401

4

41

41 xx

2

44

units 21

41

41

014110

41

e.g. (i) y

x

3xy

1-1

1

0

30

1

3 dxxdxxA

10401

4

41

41 xx

2

44

units 21

41

41

014110

41

OR using symmetry of odd function

e.g. (i) y

x

3xy

1-1

1

0

30

1

3 dxxdxxA

10401

4

41

41 xx

2

44

units 21

41

41

014110

41

OR using symmetry of odd function

1

0

32 dxxA

e.g. (i) y

x

3xy

1-1

1

0

30

1

3 dxxdxxA

10401

4

41

41 xx

2

44

units 21

41

41

014110

41

OR using symmetry of odd function

1

0

32 dxxA

104

21 x

e.g. (i) y

x

3xy

1-1

1

0

30

1

3 dxxdxxA

10401

4

41

41 xx

2

44

units 21

41

41

014110

41

OR using symmetry of odd function

1

0

32 dxxA

104

21 x

2

4

units 21

0121

(ii) y

x

21 xxxy

-2 -1

(ii) y

x

21 xxxy

-2 -1

0

1

231

2

23 2323 dxxxxdxxxxA

(ii) y

x

21 xxxy

-2 -1

0

1

231

2

23 2323 dxxxxdxxxxA1

0

2341

2

234

41

41

xxxxxx

(ii) y

x

21 xxxy

-2 -1

0

1

231

2

23 2323 dxxxxdxxxxA1

0

2341

2

234

41

41

xxxxxx

2

234234

units 21

022241111

412

(2) Area On The y axis

y

x

y = f(x)

(a,c)

(b,d)

(2) Area On The y axis

y

x

y = f(x)

(a,c)

(b,d)

(1) Make x the subjecti.e. x = g(y)

(2) Area On The y axis

y

x

y = f(x)

(a,c)

(b,d)

(1) Make x the subjecti.e. x = g(y)

(2) Substitute the y coordinates

(2) Area On The y axis

y

x

y = f(x)

(a,c)

(b,d)

(1) Make x the subjecti.e. x = g(y)

(2) Substitute the y coordinates

d

c

dyygA 3

(2) Area On The y axis

y

x

y = f(x)

(a,c)

(b,d)

(1) Make x the subjecti.e. x = g(y)

(2) Substitute the y coordinates

d

c

dyygA 3

e.g. y

x

4xy

1 2

(2) Area On The y axis

y

x

y = f(x)

(a,c)

(b,d)

(1) Make x the subjecti.e. x = g(y)

(2) Substitute the y coordinates

d

c

dyygA 3

e.g. y

x

4xy

1 2

41

yx

(2) Area On The y axis

y

x

y = f(x)

(a,c)

(b,d)

(1) Make x the subjecti.e. x = g(y)

(2) Substitute the y coordinates

d

c

dyygA 3

e.g. y

x

4xy

1 2

41

yx

16

1

41

dyyA

(2) Area On The y axis

y

x

y = f(x)

(a,c)

(b,d)

(1) Make x the subjecti.e. x = g(y)

(2) Substitute the y coordinates

d

c

dyygA 3

e.g. y

x

4xy

1 2

41

yx

16

1

41

dyyA16

1

45

54

y

(2) Area On The y axis

y

x

y = f(x)

(a,c)

(b,d)

(1) Make x the subjecti.e. x = g(y)

(2) Substitute the y coordinates

d

c

dyygA 3

e.g. y

x

4xy

1 2

41

yx

16

1

41

dyyA16

1

45

54

y

2

45

45

units 5

124

11654

(3) Area Between Two Curves

y

x

(3) Area Between Two Curves

y

x

y = f(x)…(1)

(3) Area Between Two Curves

y

x

y = f(x)…(1)y = g(x)…(2)

(3) Area Between Two Curves

y

x

y = f(x)…(1)y = g(x)…(2)

a b

(3) Area Between Two Curves

y

x

y = f(x)…(1)y = g(x)…(2)

Area = Area under (1) – Area under (2)

a b

(3) Area Between Two Curves

y

x

y = f(x)…(1)y = g(x)…(2)

Area = Area under (1) – Area under (2)

b

a

b

a

dxxgdxxf

a b

(3) Area Between Two Curves

y

x

y = f(x)…(1)y = g(x)…(2)

Area = Area under (1) – Area under (2)

b

a

b

a

dxxgdxxf

a b

b

a

dxxgxf

e.g. quadrant. positive in the

and curves ebetween th enclosed area theFind 5 xyxy

e.g. quadrant. positive in the

and curves ebetween th enclosed area theFind 5 xyxy

y

x

5xy xy

e.g. quadrant. positive in the

and curves ebetween th enclosed area theFind 5 xyxy

y

x

5xy xy

xx 5

e.g. quadrant. positive in the

and curves ebetween th enclosed area theFind 5 xyxy

y

x

5xy xy

xx 5

1or 0

010

4

5

xxxx

xx

e.g. quadrant. positive in the

and curves ebetween th enclosed area theFind 5 xyxy

y

x

5xy xy

xx 5

1or 0

010

4

5

xxxx

xx

1

0

5 dxxxA

e.g. quadrant. positive in the

and curves ebetween th enclosed area theFind 5 xyxy

y

x

5xy xy

xx 5

1or 0

010

4

5

xxxx

xx

1

0

5 dxxxA1

0

62

61

21

xx

e.g. quadrant. positive in the

and curves ebetween th enclosed area theFind 5 xyxy

y

x

5xy xy

xx 5

1or 0

010

4

5

xxxx

xx

1

0

5 dxxxA1

0

62

61

21

xx

2

62

unit 31

01611

21

2002 HSC Question 4d)

2The graphs of 4 and 4 intersect at the points 4,0 .y x y x x A

2002 HSC Question 4d)

(i) Find the coordinates of A (2) 2The graphs of 4 and 4 intersect at the points 4,0 .y x y x x A

2002 HSC Question 4d)

(i) Find the coordinates of A (2) 2The graphs of 4 and 4 intersect at the points 4,0 .y x y x x A

To find points of intersection, solve simultaneously

2002 HSC Question 4d)

(i) Find the coordinates of A (2)

24 4x x x

2The graphs of 4 and 4 intersect at the points 4,0 .y x y x x A

To find points of intersection, solve simultaneously

2002 HSC Question 4d)

(i) Find the coordinates of A (2)

24 4x x x

2The graphs of 4 and 4 intersect at the points 4,0 .y x y x x A

2 5 4 0x x

To find points of intersection, solve simultaneously

2002 HSC Question 4d)

(i) Find the coordinates of A (2)

24 4x x x

2The graphs of 4 and 4 intersect at the points 4,0 .y x y x x A

2 5 4 0x x 4 1 0x x

To find points of intersection, solve simultaneously

2002 HSC Question 4d)

(i) Find the coordinates of A (2)

24 4x x x

2The graphs of 4 and 4 intersect at the points 4,0 .y x y x x A

2 5 4 0x x 4 1 0x x

To find points of intersection, solve simultaneously

1 or 4x x

2002 HSC Question 4d)

(i) Find the coordinates of A (2)

24 4x x x

2The graphs of 4 and 4 intersect at the points 4,0 .y x y x x A

2 5 4 0x x 4 1 0x x

To find points of intersection, solve simultaneously

1 or 4x x is (1, 3)A

2(ii) Find the area of the shaded region bounded by 4 and 4.

y x xy x

(3)

2(ii) Find the area of the shaded region bounded by 4 and 4.

y x xy x

(3)

4

2

1

4 4A x x x dx

2(ii) Find the area of the shaded region bounded by 4 and 4.

y x xy x

(3)

4

2

1

4 4A x x x dx

4

2

1

5 4x x dx

2(ii) Find the area of the shaded region bounded by 4 and 4.

y x xy x

(3)

4

2

1

4 4A x x x dx

4

2

1

5 4x x dx 4

3 2

1

1 5 43 2

x x x

2(ii) Find the area of the shaded region bounded by 4 and 4.

y x xy x

(3)

4

2

1

4 4A x x x dx

4

2

1

5 4x x dx 4

3 2

1

1 5 43 2

x x x

3 2 3 21 5 1 54 4 4 4 1 1 4 13 2 3 2

2(ii) Find the area of the shaded region bounded by 4 and 4.

y x xy x

(3)

4

2

1

4 4A x x x dx

4

2

1

5 4x x dx 4

3 2

1

1 5 43 2

x x x

3 2 3 21 5 1 54 4 4 4 1 1 4 13 2 3 2

29 units2

2005 HSC Question 8b)

The shaded region in the diagram is bounded by the circle of radius 2 centred at the origin, the parabola , and the x axis.By considering the difference of two areas, find the area of the shaded region.

(3)

2 3 2y x x

2005 HSC Question 8b)

The shaded region in the diagram is bounded by the circle of radius 2 centred at the origin, the parabola , and the x axis.By considering the difference of two areas, find the area of the shaded region.

(3)

2 3 2y x x

Note: area must be broken up into two areas, due to the different boundaries.

2005 HSC Question 8b)

The shaded region in the diagram is bounded by the circle of radius 2 centred at the origin, the parabola , and the x axis.By considering the difference of two areas, find the area of the shaded region.

(3)

2 3 2y x x

Note: area must be broken up into two areas, due to the different boundaries.

2005 HSC Question 8b)

The shaded region in the diagram is bounded by the circle of radius 2 centred at the origin, the parabola , and the x axis.By considering the difference of two areas, find the area of the shaded region.

(3)

2 3 2y x x

Note: area must be broken up into two areas, due to the different boundaries.

Area between circle and parabola

2005 HSC Question 8b)

The shaded region in the diagram is bounded by the circle of radius 2 centred at the origin, the parabola , and the x axis.By considering the difference of two areas, find the area of the shaded region.

(3)

2 3 2y x x

Note: area must be broken up into two areas, due to the different boundaries.

Area between circle and parabola and area between circle and x axis

It is easier to subtract the area under the parabola from the quadrant.

It is easier to subtract the area under the parabola from the quadrant.

1

2 2

0

1 2 3 24

A x x dx

It is easier to subtract the area under the parabola from the quadrant.

1

2 2

0

1 2 3 24

A x x dx 1

3 2

0

1 3 23 2

x x x

It is easier to subtract the area under the parabola from the quadrant.

1

2 2

0

1 2 3 24

A x x dx 1

3 2

0

1 3 23 2

x x x

3 21 31 1 2 1 03 2

It is easier to subtract the area under the parabola from the quadrant.

1

2 2

0

1 2 3 24

A x x dx 1

3 2

0

1 3 23 2

x x x

3 21 31 1 2 1 03 2

25 units6

Exercise 11E; 2bceh, 3bd, 4bd, 5bd, 7begj, 8d, 9a, 11, 18*

Exercise 11F; 1bdeh, 4bd, 7d, 10, 11b, 13, 15*