Download - 11X1 T17 04 areas
![Page 1: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/1.jpg)
Areas(1) Area Below x axis
y = f(x)y
x
![Page 2: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/2.jpg)
Areas(1) Area Below x axis
y
x
y = f(x)
a b
1A
![Page 3: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/3.jpg)
Areas(1) Area Below x axis
y
x
y = f(x)
a b
1A
b
adxxf 0
![Page 4: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/4.jpg)
Areas(1) Area Below x axis
y
x
y = f(x)
a b
1A
b
adxxf 0
b
adxxfA1
![Page 5: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/5.jpg)
Areas(1) Area Below x axis
y
x
y = f(x)
a b
1A
b
adxxf 0
b
adxxfA1
c2A
![Page 6: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/6.jpg)
Areas(1) Area Below x axis
y
x
y = f(x)
a b
1A
b
adxxf 0
b
adxxfA1
c2A
c
bdxxf 0
![Page 7: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/7.jpg)
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
![Page 8: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/8.jpg)
Area below x axis is given by;
![Page 9: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/9.jpg)
Area below x axis is given by;
c
bdxxfA
![Page 10: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/10.jpg)
Area below x axis is given by;
c
bdxxfA
OR
c
bdxxf
![Page 11: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/11.jpg)
Area below x axis is given by;
c
bdxxfA
OR
c
bdxxf
OR
b
cdxxf
![Page 12: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/12.jpg)
e.g. (i) y
x
3xy
1-1
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e.g. (i) y
x
3xy
1-1
1
0
30
1
3 dxxdxxA
![Page 14: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/14.jpg)
e.g. (i) y
x
3xy
1-1
1
0
30
1
3 dxxdxxA
10401
4
41
41 xx
![Page 15: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/15.jpg)
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
![Page 16: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/16.jpg)
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
![Page 17: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/17.jpg)
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
![Page 18: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/18.jpg)
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
![Page 19: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/19.jpg)
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
![Page 20: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/20.jpg)
(ii) y
x
21 xxxy
-2 -1
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(ii) y
x
21 xxxy
-2 -1
0
1
231
2
23 2323 dxxxxdxxxxA
![Page 22: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/22.jpg)
(ii) y
x
21 xxxy
-2 -1
0
1
231
2
23 2323 dxxxxdxxxxA1
0
2341
2
234
41
41
xxxxxx
![Page 23: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/23.jpg)
(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
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(2) Area On The y axis
y
x
y = f(x)
(a,c)
(b,d)
![Page 25: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/25.jpg)
(2) Area On The y axis
y
x
y = f(x)
(a,c)
(b,d)
(1) Make x the subjecti.e. x = g(y)
![Page 26: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/26.jpg)
(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
![Page 27: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/27.jpg)
(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
![Page 28: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/28.jpg)
(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
![Page 29: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/29.jpg)
(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
![Page 30: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/30.jpg)
(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
![Page 31: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/31.jpg)
(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
![Page 32: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/32.jpg)
(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
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(3) Area Between Two Curves
y
x
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(3) Area Between Two Curves
y
x
y = f(x)…(1)
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(3) Area Between Two Curves
y
x
y = f(x)…(1)y = g(x)…(2)
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(3) Area Between Two Curves
y
x
y = f(x)…(1)y = g(x)…(2)
a b
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(3) Area Between Two Curves
y
x
y = f(x)…(1)y = g(x)…(2)
Area = Area under (1) – Area under (2)
a b
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(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
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(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
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e.g. quadrant. positive in the
and curves ebetween th enclosed area theFind 5 xyxy
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e.g. quadrant. positive in the
and curves ebetween th enclosed area theFind 5 xyxy
y
x
5xy xy
![Page 42: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/42.jpg)
e.g. quadrant. positive in the
and curves ebetween th enclosed area theFind 5 xyxy
y
x
5xy xy
xx 5
![Page 43: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/43.jpg)
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
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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
![Page 45: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/45.jpg)
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
![Page 46: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/46.jpg)
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
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2002 HSC Question 4d)
2The graphs of 4 and 4 intersect at the points 4,0 .y x y x x A
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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
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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
![Page 50: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/50.jpg)
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
![Page 51: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/51.jpg)
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
![Page 52: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/52.jpg)
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
![Page 53: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/53.jpg)
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
![Page 54: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/54.jpg)
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
![Page 55: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/55.jpg)
2(ii) Find the area of the shaded region bounded by 4 and 4.
y x xy x
(3)
![Page 56: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/56.jpg)
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
![Page 57: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/57.jpg)
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
![Page 58: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/58.jpg)
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
![Page 59: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/59.jpg)
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
![Page 60: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/60.jpg)
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
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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
![Page 62: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/62.jpg)
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.
![Page 63: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/63.jpg)
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.
![Page 64: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/64.jpg)
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
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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
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![Page 67: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/67.jpg)
It is easier to subtract the area under the parabola from the quadrant.
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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
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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
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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
![Page 71: 11X1 T17 04 areas](https://reader034.vdocument.in/reader034/viewer/2022052505/55640ad3d8b42aa9518b54f6/html5/thumbnails/71.jpg)
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
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Exercise 11E; 2bceh, 3bd, 4bd, 5bd, 7begj, 8d, 9a, 11, 18*
Exercise 11F; 1bdeh, 4bd, 7d, 10, 11b, 13, 15*