Download - 11X1 T17 04 areas
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*