SPM Add Math Form 5 Chapter 3 Integration

SPM Add Math Form 5 Chapter 3 Integration CHAPTER 3 : INTEGRATION INDEFINITE INTEGRAL Integration as the reverse process of differentiation 𝑑𝑦 = 3𝑥. 𝑑𝑥 𝑑 1 that 𝑑𝑥 2 𝑥 2

1) (a) Find y if (b) Given

+ 𝑐 = 𝑥 , where c is a constant, find

𝑥 𝑑𝑥.

Integral of axn 2) Integrate the following with respect to x . (a) x5 (b) 2x3 (c) -3x2

(d) -2x

(e) 7

(f) -10

3) Integrate the following with respect to x . (a)

4) If

1

(b)

𝑥4

𝑑𝑦 𝑑𝑥

=

4 7𝑥 5

,

2 3

𝑥3

−8

(c)

(d)

𝑥2

2 5𝑥 3

(e)

−2

(f)

𝑥7

−4 5

find y.

Integrals of algebraic expressions 5) Find the following indefinite integrals. (a)

𝑥 2 + 1 𝑑𝑥

(c)

(b)

𝑥 𝑥 − 3 𝑑𝑥

(d)

6) Find

1

𝑥 3 2 + 𝑥 6 𝑑𝑥.

𝑥 2 −2 𝑥2

𝑑𝑥

𝑥−1

2

(e) 𝑑𝑥

(f)

𝑥 − 3 𝑥 + 5 𝑑𝑥 6𝑥 3 +3 𝑥2

𝑑𝑥

[SPM 1998 P2 clone]

Finding the constant of integration 7) If

𝑑𝑦 𝑑𝑥

= 3𝑥 2 − 𝑥 − 5 and y=1 when x=2, find the constant of integration, c.

Equation of curves from functions of gradients 8) Find the equation of a curve whose gradient is 2x – 5 and passes through the point (1,2). 9) Find the equation of the curve which passes through the point (2,-4) and for which

𝑑𝑦 𝑑𝑥

= 𝑥 − 3𝑥 2 .

[SPM 1998 P2 clone]

10) Given that

𝑑𝑦 𝑑𝑥

= 3𝑥 2 − 4𝑥 + 3 and y=4 when x=2, find y in terms of x. [SPM 2001 P2 clone]

11) A curve has a gradient function kx2+x where k is a constant. The tangent to the curve at 1 2

the point (1, ) is parallel to the straight line y=4x-7. Find (a) the value of k, (b) the equation of the curve. Copyright www.epitomeofsuccess.com

[SPM 2005 P2 clone]

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SPM Add Math Form 5 Chapter 3 Integration Integration of (ax+b)n 12) Integrate the following with respect to x . (a) (2x+6)5

(b)

1 (3𝑥−1)

(c) 4(5 – x)3

4

13) Given that the gradient of a curve is

8 (2𝑥−3)3

(d)

2 (2𝑥−3)6

and that the curve passes through the point (2,-5).

Find the equation of the curve. 2 (𝑥+1)5

14) Given that

𝑑𝑥 = 𝑘(𝑥 + 1)𝑛 + 𝑐, find the values of k and n.

[SPM 2003 P2 clone]

DEFINITE INTEGRAL Definite integrals of algebraic expressions 15) Evaluate (a)

4

2 2𝑥 3 𝑑𝑥 0

8

(b) 2 (𝑥 3 − 2 ) 𝑑𝑥 𝑥

2

(c) −2

𝑥 4 +6 𝑥4

𝑑𝑥

(d)

6 3

𝑥 − 4 (𝑥 + 6) 𝑑𝑥

16) Evaluate (a)

2 (3𝑥 1

𝑑 𝑑𝑥

17) Given that

𝑓 𝑥 3𝑥−1 𝑥2

18) Given that 𝑦 = 1 −1

𝑑

𝑥

𝑑𝑥

1−2𝑥

3 𝑓 1

20) Given that 1 𝑓 3

= 3𝑥 2 − 1, find the value of 𝑓 3 − 𝑓 2 . and

𝑑𝑦 𝑑𝑥

= 3𝑔 𝑥

2 4

=

1 (1−2𝑥)2

𝑥 𝑑𝑥 = 4 and

𝑥 𝑑𝑥

21) Given that (a)

1

such that g(x) is a function of x. Find the value of

𝑔 𝑥 𝑑𝑥.

19) Show that

(a)

5

(b) 1 𝑑𝑥 (2𝑥−1)2

− 4)3 𝑑𝑥

(b) 4 𝑕 2

5 3

. Hence or otherwise, evaluate 5 𝑓 3

3 1 ( )2 1 1−2𝑥

𝑑𝑥.

𝑥 𝑑𝑥 = 3, find

2𝑓 𝑥 𝑑𝑥

(c)

5 𝑓 1

𝑥 𝑑𝑥

(d)

5 1

3𝑓 𝑥 + 𝑥 𝑑𝑥

𝑥 𝑑𝑥 = 5, evaluate

𝑕 𝑥 − 𝑥 𝑑𝑥 ,

(b) the value of k for which 22) Evaluate 23) Given that 24) Given that

1 3 −1 𝑥 4

𝑕 𝑥 + 𝑘𝑥 3 𝑑𝑥 = 35.

+ 8𝑥 𝑑𝑥 .

𝑑

𝑥3

𝑑𝑥

2𝑥−5

𝑘 1

4 2

= 𝑓(𝑥), calculate the value of

[SPM 2001 P2 clone] 2 0

𝑓 𝑥 + 𝑥 𝑑𝑥.

4𝑥 − 3 𝑑𝑥 = 3, where k>0, find the value of k.

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[SPM 2002 P1 clone] [SPM 2004 P1 clone]

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SPM Add Math Form 5 Chapter 3 Integration 25) Given that

3 𝑓 1

𝑥 𝑑𝑥 = 5 and

26) Given that

6 𝑕 2

𝑥 𝑑𝑥 = 8, find

(a) the value of

2 𝑕 6

(b) the value of p if

3 1

𝑓 𝑥 + 𝑘𝑥 2 𝑑𝑥 = 31, find the value of k. [SPM 2005 P1 clone]

𝑥 𝑑𝑥 , 6 2

𝑝𝑥 − 𝑕 𝑥 𝑑𝑥 = 24.

[SPM 2006 P1 clone]

Areas under curves using formula 27) For each of the following diagrams, find the area of the shaded region. (a) (b)

4

28) The diagram shows part of the curve 𝑦 = 2 − 1. 𝑥 Find the area of the shaded region.

2

29) The diagram shows part of the curve 𝑦 = 2 𝑥 and of the line y=2x intersecting at A. Find (a) the coordinates of A, (b) the area of the shaded region.

30) For each of the following diagrams, find the area of the shaded region. (a) (b)

31) The diagram shows the shaded region, enclosed by the curve 𝑥 = 𝑦(𝑦 − 3) and the y- axis. Calculate the area of the shaded region.

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SPM Add Math Form 5 Chapter 3 Integration 32) The diagram shows part of the curve 𝑦 = 𝑥 2 And of the line y=6-x intersecting at point P. Find (a) the coordinates of P, (b) the area of the shaded region.

33) The diagram shows part of the curve 𝑦 = 𝑥 2 − 5𝑥 + 8 and part of the line y=x. Find (a) the coordinates of the points P and Q, (b) the area of the shaded region.

34) The diagram shows part of the curve 𝑦 2 = 9𝑥 and of the line x=1, intersecting at the points P and Q. Find (a) the coordinates of P and Q, (b) the area of the shaded region.

35) The diagram shows part of the curve 𝑦 = 𝑓(𝑥) which touches the x-axis at A and cuts the y-axis at B. The tangent to the curve at the point B and the x-axis are parallel. Given that 𝑓 ′ 𝑥 = −3𝑥 𝑥 + 2 , find (a) the coordinates of A, (b) 𝑓 𝑥 , (c) the coordinates of B, (d) the area of the shaded region. [SPM 1999 P1 clone]

36) The diagram shows a straight line 𝑦 = 𝑓(𝑥) which intersects the curve 𝑦 = (𝑥 + 4)2 0 at the point (-1,9). Given that −1 𝑓 𝑥 𝑑𝑥 = 7, find the area of the shaded region. [SPM 2001 P1 clone]

37) The diagram shows part of the curve 𝑦 = 2𝑥 2 and the straight line x=k. If the area of the shaded region is 18 unit2, find the value of k. [SPM 2003 P1 clone]

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SPM Add Math Form 5 Chapter 3 Integration 38) The diagram 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 8 unit2, find the value of 𝑏

3𝑓 𝑥 𝑑𝑥. 𝑎

[SPM 2006 P1 clone]

Volumes of revolutions using formula 39) For each of the following shaded regions, find the volume of the solid generated by rotating 360° about the x-axis. (a) (b)

40) For each of the following shaded regions, find the volume of the solid generated by rotating 360° about the y-axis. (a) (b)

2

41) The diagram shows part of the curve 𝑦 = 𝑥 and of the line y=x+1 intersecting at P. Find (a) the coordinates of P, (b) the volume of solid formed by rotating the shaded region 360° about the x-axis.

42) The diagram shows part of the two curves 𝑦 = 𝑥 2 and 𝑦 = 8 − 𝑥 2 intersecting at A. Find (a) the coordinates of A, (b) the volume generated when the shaded region is rotated through 360° about the y-axis.

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SPM Add Math Form 5 Chapter 3 Integration 43) The diagram shows the curve 𝑦 = 𝑥 2 − 4 and 𝑥 𝑦 the line − = 1. Find the volume generated 4

6

when the shaded region is rotated 360° about the y-axis. [SPM 1998 P1 clone]

44) The diagram shows part of the curve 𝑦 = 𝑥 2 + 4. The shaded region is bounded by the curve, the y-axis and the line y=k. If the volume of revolution obtained by rotating the shaded region through 360° about the y-axis is 2𝜋 unit3, evaluate k. [SPM 2001 P1 clone]

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