Chapter 3 Integration - Area

Chapter 3 Integration Additional Mathematics Form 5

Determine equations of curves from functions of gradient

1) Find the equation of the curve with gradient function 4 x−1and passes through the point (1,6).

2) Find the equation of the curve with gradient function

x2−4 x and passes through the point (4,0).


x2( x−3 )and passes through the point (2,-6).


(2 x+1 )3and passes through the point (

12 , -3).

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3 x2−4 and passes through the point (-1,6).


x2(2 x+1)and passes through the point (1,-1).

7) Find the equation of the curve with gradient function 3 x2−6 x and passes through the point A(1,-12) 8) Given that

dydx

=2x+2 and y=6 when x=-1, express y

in terms of x. (SPM03/2/3a)

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9) Find the equation of the curve for which

dydx

= 4

( x+2)2

and which passes through the point (2,7).10) A curve is such that

dydx

=(3 x−2 )2

. Given that the curve passes through (1,2), find its equation.

11) The gradient of a curve, at the point (x, y) on the curve, is

given by

x2−4x2

. Given that the curve passes through the point (2,7), find the equation of the curve.

12) The gradient at any point on a particular curve is given by

the expression x2+16

x2, where x>0. Given that the

curve passes through the point P(4,18), find the equation of the curve.

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13) Find the equation of the curve which passes through the

point (3,6) and for which

dydx

=2x ( x−3 ).

14) The gradient at any point (x, y) on a curve is given by 6 x2+6 x−5 . Given that the curve passes through the point (2, 12), find the equation of the curve.

15) A curve for which

dydx

=2x−5, passes through the point

P(4,-2). Find the equation of the curve.16) A curve for which

dydx

=ax−6, where a is a constant,

passes through the point (2,1). At this point the gradient of the curve is 4. Find(a) the value of a,(b) the equation of the curve.

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dx 3 X 4^

, 1 , 3 =

ALPHA

dx ( X +X2

) 2,4 =

ALPHA 13

^ 0 ,


Kaedah ALTERNATIF UNTUK DEFINITE INTEGRAL

(kiraan langsung melalui kalkulator, untuk mendapatkan nilai kamiran tanpa kaedah penggantian had atas dan had bawah)

1 Pilih Contoh 1:

Contoh 2:

2Taipkan persamaan SOALAN ( pilih ALPHA untuk fungsi x)

3 Masukkan nilai had BAWAH,

4 Masukkan nilai had ATAS

5 =

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Jawapan: 145.2

Jawapan: 6962.5

NOTA : Kalkulator mungkin mengambil sedikit masa untuk proses pengiraan

(a)

(b) =

(c)


Definite Integrals

1. Given that , find

(a)

(b) the value of k if


(a)

(b)

3. Given that , find the value of m if 4. Given that , find the value of k if

.

5. Given that and , where k is a constant and k > -2. Find

(a)(b) The value of k.


(a)

(b) The value of k where

7. Given that and , find the value of w.

8. Given that , find the value of k if

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(a)

(b)

10. Given that and

, find the value of h.


(a)

(b)

12. Given that ,find the value of k if

13. Given that , find the value of

.

14. Given that and . Find the

value of


(a)

(b) The value of k if


(a)

(b)

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value of k where k is a constant if


value of m if .

19. Given that and , find the value of

(a)

(b) k if

20. Given that and . Find

.


.


.

23. Given that , find 24. Given that and , find the value of k.

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25. Given that and , find the value of k.


.

27. Given that and . Find the value of

(a)

(b) h if

28. Given that and , where f(x) is a

function of x. Evaluate


(a) +

(b) k if


(a)

(b) h if

31. Given that and . Find the value of

(a)

(b) k if

32. Given that and . Find

(a)

(b)

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33. Given that and .Find the value of k.

34. Given that . Find the value of p if


(a)

(b)

36. Given that , find .

37. Given that , find . 38. Given that , find

(a) the value of

(b) the value of k if

39. Given that , find the value of 40. Given that ,find the value of

.

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x

y

O a b

y = f(x)

Diagram 8 x

y

O a b

y = g(x)

Diagram 9

x

y

O

y = f(x)

5

4

Diagram 10

x

y

O

y = f(x)

6

8

Diagram 11


3.4 Area under a curve1. (SPM 06, P1,Q20)

Diagram 8 shows the curve y = f(x) cutting the x-axis at x = a and x = b.

Given that the area of the shaded region is 5 unit2, find

the value of .

Answer :

2. Diagram 9 shows the curve y = f(x) cutting the x-axis at x = a and x = b.

Given that the area of the shaded region is 6 unit2, find the

value of .

Answer :

3. Diagram 10 shows part of the curve y = f(x).

Given that = 15 unit2, find the area of the shaded region.

Answer :

4. Diagram 11 shows part of the curve y = f(x).

Given that the area of the shaded region is 40 unit2, find

the value of .

Answer :

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x

y

O

2

(4, 8)

Diagram 12

●

x

y

O

10

(6, 2)Diagram 13

●

y

O x

O

y

x

P

x = 1

Q

y2 = 4(x + 1)

(h, 3)

y

Ox

y = 2x + 1

y = (x 2)(x 4)

y = x(x 1)(x + 3)

y

xO


5. (SPM 01) Diagram 12 shows the sketch of part of a curve.

(a) Shade, on the given diagram, the region

represented by .

(b) Find the value of Answer :

6. Diagram 13 shows the sketch of part of a curve.

(a) Shade, on the given diagram, the region represented by

.

(b) If = p , find , in terms of p, the value of

.

Answer :

Find the area of the shaded region

1.

Diagram shows the graph y = x2 4x + 7 and y = 7 x.Find the area of the shaded region.

[1994/K2/5] [5 marks]

2.

Diagram shows part of the curve y2 = 4(x + 1) that intersect the straight line x = 1 at point Q. Given the straight line PQ is parallel to the x-axis. Find the area of the shaded region.

[1995/K1/18] [4 marks]

3. 4.

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Find the area bounded by the curve, the x-axis, the straight line x = 2 and x = 1.

[1996/K1/18][6 marks]

Find the area of the shaded region.[1997/K1/18] ][7marks]

Find the area of the shaded region

1. 2.

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3. 4.

5. 6.

7. 8.

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9. 10.

11. 12.

Volume of Revolution

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1.

Find the volume of revolution, in terms of , when the area bounded by the curve, the y-axis and the line y=5

is rotated through about the y-axis.

2.

Find the volume generated, in term of , when the region enclosed by the curve, the y-axis and the straight

line y=2 is rotated through about the y-axis.

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3.

Find the volume generated, in term of , when the region bounded by the curve, the y-axis and the straight

line y=3 is rotated through about the y-axis.

4.

Find the volume generated in terms of π , when the

region bounded by the curve,the y -axis and y = 6

is revolved through 360° about the y -axis.

5.

A region is bounded by the curve, the x-axis and the straight line x = -2 and x = -3 . The region is

revolved through about the x-axis. Find the volume of generated, in terms of ,

[Ans : ]

6.

Find the volume of revolution, in terms of , when

the shaded region Q is rotated through about the x-axis.

[Ans : ]

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y

xO

y = x2 + 2


7.

Find the volume of revolution, in terms of , when the region bounded by the curve, and the straight line is

rotated through about the y-axis.

[Ans : ]

8.

Find the volume of revolution, in terms of , when the

shaded region A is rotated through about the y-axis.

9.

Find the volume of revolution, in terms of , when the region under the curve and the x-axis is rotated

through about the x-axis.

10.

Find the volume of revolution, in terms of , when the region bounded by the curve, the x-axis, the y-axis and the straight line x = 2 is revolved through 360° about the x-axis.

[Ans : ]

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11.

Find the volume generated, in terms of , when the shaded region B is revolved through 360o about the x-axis.

[Ans : ]

12.

Find the volume of revolution, in terms of , when the region bounded by the curve, the x-axis, the y-axis is revolved through 360° about the x-axis.

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Chapter 3 Integration - Area

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