11 ALGEBRA

EXAMPLE 95

Solution

EXAMPLE 96

Solution

Find the length L of the graph of f(x) = 4(x – 1)3/2 between x = 1 and x = 2.

Using the formula we get:

f (x)= 6 (x – 1 )1/ 2

	2	1+(6(x 1)12 )2					2						2
L =							dx=	1+36( x 1) dx=						36 x 35 dx
	1						1						1
u = 36x – 35 du = 36 dx or dx= du
											36
	1	1		du = u	3	2	= (36x 35)		3	2	2	37	37 1.
dx =		u	2								|=
	36
				54				54			1		54

Find the circumference of a circle with radius 2 units.

Let us assume that the center of the circle is at the origin of a graph, then the equation of the circle is x2 + y2 = 4.

So y =

4 x2 .

Now let us divide the circle into four parts and find the length of just one part:

2

L =

1+((

4 x2 ) )2

dx.

0

2

x

So the circumference of the circle = 4

1+(

)2

dx

0

4 x2

2

x

2

y

= 4

1+

4

2 dx

y = ±

0

x

2

4

= 4

dx

4 x

2

0

2

dx

2

x

= 8

4 x2

0

= 8 (arcsin

x

2

x2 + y2 = 4

2

) |

0

= 8 (arcsin1– arcsin0)

= 4 units.

Integrals

69

3.Calculating the Area of a Surface of Revolution (optional)

FINDING SURFACE OF REVOLUTION

If a function f(x) has a continuous first derivative on [a, b] then the area A of the surface generated by revolving the curve about x-axis is

			b	1+ f (x) 2 dx.
		A = 2 f (x)
			a
		Such a surface is called a surface of revolution.
	97 Find the surface area of a sphere with radius r = 3 cm.
EXAMPLE
Solution Let us take the circle x2 + y2 = 9			y =	9 x2 .

Now let us use the arc between x = 0 and x = 3 and rotate it. This will give us half of the surface area of the sphere, so we need to multiply the result by 2 to obtain the whole surface area.

	3									y
surface area = 2 2		9 x2		1+((				9 x2 ) )2	dx
	0								3	x2 + y2 = 9
3						2
= 4	9 x2 1+				9	x		2 dx
0						x
3		2		3					–3	3	x
= 4							dx
	9 x			9 x2
0
		3
= 4 (3x) |									–3
		0
= 36 cm2

Check Yourself 18

1.Find the length of the curve y = 2(x + 3 )3/2 between x = 1 and x = 3.

2.Find the length of the curve y = 4x2 – x + 1 on the interval [0, 1]

3.Find the length of the curve y = x3/2 on the interval [0, 2].

4. Find the area of the surface of revolution which is generated by rotating the curve y = 2x + 1 about the x-axis on the interval [1, 3].

5.A parabolic reflector is obtained by rotating the parabola y = ñx on the interval [1, 2] about the x-axis. What is the surface area of the reflector?

Answers

1.	110 55 74 37	2.	9 2 ln(	2 1)	3.	22 22 8	4. 20 ñ5	5.	9		5 5
	27		4			27			2		6

70	Algebra 11

PRACTICAL INTEGRAL APPLICATIONS

We have seen how to use the definite integral to find the

area under a curve, the volume of a solid, and the length of

a curve. These results have many practical applications.

For example, Table 1 shows a graph about a cell phone

company. The graph shows the number of new users the

company hopes to have per month. How many users will

there be after five months? The

answer is the area under the

graph.

The definite integral is also

number of users (v)

f(x)

useful in economics and busi-

ness. Statistics is the branch of

3000

mathematics that studies and

2000

processes data. Statisticians use

tables and graphs to find out

1 2 3

months

about changes over time, for

Table 1

example, changes in a compa-

ny’s income, or changes in the population of a city or

country. For example, imagine you are studying the pop-

ulation of an island. You have found that the popula-

tion increase P over t years is given by P = 25ñt + 20. The current population is

1200. How many people will be living on the island in thirty years’ time? (This

problem is left as an exercise for you. Hint: use the definite

integral.)

The definite integral also has applications in circuit design,

architecture, astronomy and many other fields. Integrals

tell us about the dilation of electronic circuits, the curves

and surface areas of buildings, and the movements of the

stars and planets.

EXERCISES 1.3

A.Finding the Area Under a Curve

1.Find the area of the region bounded by the graph of y = 4 – x2 and the x-axis.

2.Find the area of the region bounded by the line y = 3 + x and the x and y axes.

3.Find the area of the region bounded by y = 2x – 1 and the x-axis on the interval [0, 3].

4.Find the area of the region bounded by the graph of y = 1 – 3x and the x-axis on the interval [2, 5].

5.Find the area of the region bounded by the graph of y = x2 + 5 and the x-axis on the interval [0, 3].

6. Find the area of the region bounded by the graph of y = x2 – 3x – 4 and the x-axis on the interval [–1, 7].

7.Find the area of the region bounded by the graphs of y = x – 1, y = 0, x = 1 and x = 3.

8.Find the area of the region bounded by the graphs of y = x3 – 1, y = 0, x = 1 and x = 2.

9.Find the area of the region bounded by the graphs of y = 3 – x2, y = 0, x = 0 and x = 2.

10.Find the area of the region bounded by the graphs of y = 2x + 1, x = 0, y = 1 and y = 3.

11.Find the area of the region bounded by the graph of y = 3x – 1, the y-axis, and the lines y = 0 and y = 2.

12.Find the area of the region bounded by the graph of y = ñx, the y-axis, and the lines y = 1 and y = 3.

13.Find the area bounded by the graph of x = y2 – 4 and the y-axis.

14.Find the area of the region bounded by the graph of x = y2 – 3y + 2 and the y-axis.

15.Find the area of the region bounded by the graphs of y = 2x2 – 3x + 1 and y = 3 on the interval [2, 3].

16.Find the area of the region bounded by the graphs of y = 2x – 5, y = –2, x = 1 and x = 3.

17.Find the area of the region bounded by the curves y = x2 – 1 and y = 1 – x2.

18.Find the area of the region bounded by the curves f(x) = x3 and g(x) = ñx.

19.Find the area of the region bounded by the curves y = 4 – x2 and y = x2 + 2.

72	Algebra 11

20.Find the area of the region bounded by the graphs of y = 2x2 – 3x + 1 and y = –8x + 4.

21.Find the area of the region bounded by the graphs of y = x2 – 1 and y = 3x +3.

22.Find the area of the region bounded by the graphs of y = 4 – x2 and y = 2x + 1.

23.Find the area of the region bounded by the graphs  of y = x3 – 2x2 and y = 3x.

24.Find the area of the region bounded by the graphs of f(x) = x2 and g(x) = 4.

25.Find the area of the region bounded by the graphs of f(x) = 3 – x2 and g(x) = 2.

26.Find the area of the region bounded by the graphs of y = x2 and x = y2.

27.Find the area of the region bounded by the graphs of y = cos x and y = sin x on the interval [0, ]

28.Find the area of the region bounded by the graphs

	 of y = sin x and y = cos 2x on [0,		].
		3

29. Find the area of the region bounded by the graphs

of y =	cos x	and y =	sin x	on [		,		].
					3		2
3		3

30.Find the area of the region bounded by the graphs of y = 2sin x and y = 3cos x on [0, 6 ].

31.Find the area of the region bounded by the curve y = sin x and the x-axis on [0, 2 ].

32.Find the area of the region between the graph of y = 5 cos4x and the x-axis on [0, 4].

33.Find the area of the region bounded the graph of

y = 3 sin x, the x-axis, and the lines x = 0 and x 3.

34. What is the area of the region bounded by the  graphs of y = x2, y = 3x2 and y = 4x?

35. In the figure the shaded	y
area is 12 cm2 and
2
f (x) dx = 0.		2		x
	–2
2
2
What is f (x) dx?			f(x)
0

36.The area of the region bounded by y = ax2 (a > 0), the x-axis and the line x = 3 is 18 cm2. What is the value of a?

Integrals

73

37. The figure shows the graph of the function f(x).

1				5
f (x) dx = –5 and				f (x) dx = 5 are given. Find
2				3
the total area of the shaded region.
		y
	–3	–2		1	5	x

38. The figure shows the y

graph of f(x).
7			1	2	7
If f (x) dx = –3	and					x
					B
1
the area B is 8 cm2,
7
what is | f (x)| dx?
1

39.Find the area of the region bounded by the graphs of y = x2 + 3x – 1, y = 0, x = –3 and x = 0.

40.Find the area of the region bounded by the graphs of y = x2 and y = 1 on the interval [–2, 1].

41.Find the area of the region bounded by the graphs  of y = x3 + 1 and y = 5 on the interval [0, 2].

42.Find the area of the region bounded by the graphs of y = 1 + 3x, y = 8, x = 2 and x = 3.

43.Find the area of the region bounded by the curves y = 2x2 – 3x + 5 and y = 10 – x – x2.

44.Find the area of the region bounded by the curves  y = x3 + x2 + 2x and y = 7x2 – 9x + 6.

45.Find the area of the region bounded by the graphs  of x = y2 and y = x – 3.

46.Find the area of the region bounded by the graphs

	 of y = sin3x, y = 2cos x, x = 0 and x =		.
		2

47. Find the area of the region bounded by the graphs

	5		7 ].
of y = cos 2x and y = 2sin x on [		,
	3		4

48. Find the area of the region bounded by y2 = x,  y = 81 x2 , y = 1, y = 23.

49. Find the area of the	y
 shaded region in the	y = x2
figure.	2

y = 1x

x

2

74	Algebra 11

B.Other Applications

50.Find the volume of the solid figure generated by

rotating the area of the region bounded by y = 2x + 5, x = 2 and x = 3 around the x-axis.

51. Find the volume of the solid figure generated by rotating the area of the region bounded by y = x2 + 1 and the x-axis on [0, 1] about the x-axis.

52. Find the volume of the solid figure generated by rotating the area of the region bounded by y = x3 – 1 and the x-axis on [1, 2] around the x-axis.

53. Find the volume of the solid figure generated by rotating the area of the region bounded by y = 3x + 1, the x-axis, and the lines x = 1 and x = 3 about the x-axis.

54. Find the volume of the solid figure generated by rotating the area of the region bounded by y = 1 – x2 and the x-axis around the x-axis.

55. Find the volume of the solid figure generated by rotating the area of the region bounded by y = x2 – 4 and the x-axis about the x-axis.

56. Find the volume of the solid figure generated by rotating the area of the region bounded by y = 2x – 1, the y-axis, and the lines y =1, y = 2 0 about the x-axis.

57. Find the volume of the solid figure generated by rotating the area of the region bounded by y = 2x2 – 1, the y-axis, and the lines y = 0 and y = 3 about the y-axis.

58. Find the volume of the solid figure generated by rotating the area of the region bounded by y = x2 + 4 and y = 2 on the interval [1, 3] about the x-axis.

59. Find the volume of the solid figure generated by  rotating the area of the region bounded by y = x2 – 4 and y = 3x + 6 and x-axis about the

x-axis.

60. Find the volume of the solid figure generated by rotating the area of the region bounded by y = 4x – 1, and the x-axis on [0, 3] about y-axis.

61. Find the volume of the solid figure generated by  rotating the area of the region bounded by y = sin x, x = 0, x = and x-axis about the

x-axis.

62. Find the volume of the solid figure generated by rotating the area of the region bounded by y = cos 2x, x 2 , and x-axis about the x-axis.

63.Find the volume of the solid figure generated by rotating the area of the region bounded by y = x2 and y = x about the x-axis.

64.Find the volume of the solid figure generated by rotating the area of the region bounded by y = x2 and y = x about the y-axis.

65.Find the volume of the solid figure generated by

rotating the area of the region bounded by y2 = x + 4, x = 2 and y = 2 about the x-axis.

66. Find the volume of the solid figure generated by rotating the area of the region bounded by y = 2x2 + 3x – 1 and y = x2 + x – 2 about the x-axis on [1, 3].

67.Find the volume of a cone with radius r = 3 cm and altitude 4 cm by using integration.

Integrals

75

68. Find the volume of the solid figure generated by rotating the area of the region bounded by y = x2 + 1 and y = 3x – 1 about the x-axis.

69.Find the volume of the solid figure generated by rotating the area of the region between y = tan x

and the x-axis on the interval 0,				about the
x-axis through 180°.		3

70. Find the volume of the solid figure generated by rotating the area of the region bounded by y = 5 – x2 and y = x2 + 3 about the x-axis.

71.Find the volume of the solid figure generated by rotating the area of the region bounded by f(x) = –x2 and g(x) = x2 – 3 about the x-axis.

72.Find the volume of the solid figure generated by

rotating the area of the region bounded by y = x2 + x + 1, x = 1, x = 2 and the x-axis about the x-axis through 90°.

73. Find the volume of the solid figure generated by  rotating the area of the region bounded by y = 5 – x2 , y =2, x = 0, x = 1 about y = 1 .

74. Find the volume of the solid figure generated by rotating the area of the region bounded by y = 1 – x2, the x-axis, x = 1 and x = 3 about the y-axis.

75. Find the volume of the solid figure generated by rotating the area of the region bounded by y = x3, the x-axis, x = 0 and x = 2 about the y- axis.

76.Find the length of the graph y = 3x + 1 between

x= 0 and x = 4.

77.Find the length of the curve y = 2 (x – 1)3/2 on the interval [1, 2].

78.Use integration to find the circumference of a circle with radius 5 cm.

79.Find the length of the parabola y2 = x on the interval [0, 1].

80.Find the length of the graph y = x3/2 over [0, 1].

81.Find the length of the curve y = x2 – 1 between



x= 0 and x = 1.

82.Use integration to find the surface area of a sphere with radius 2 cm.

83.Find the surface area of the solid figure generated by revolving the parabola y = x2 around the x-axis on the interval [0, 1].

84.Find the surface area of the solid generated by

rotating the curve y =	2x3 / 2	on [1, 2] about the
	3

x-axis.

85.Calculate the surface area of the solid obtained by rotating the graph of f(x) = ñx on the interval [0, 1] about the y-axis.

76	Algebra 11

CHAPTER REVIEW TEST 1A

1. If f (x)= (x2		x+3) dx		then what is f (2)?
A) 1	B) 3	C)	8	D)	23	E) 5
			3		3

2. (3x2 +4x 5) dx =?
A) x3 + x2 – 5x	B) x3 + 2x2 – 5x + c
C) 3x3 + 4x2 – 5x + c	D) 6x – 4 + c

E) 3x3 – 4x2 – 5x + c

3. f (x)= x+ x		x x2	dx is given. Find f(4) if the
		x
constant of the integration is 0.
A) 1	B) 2	C)		9	D)	17		E)	8
				2
							3		15

4.What is the primitive of the function f(x) = 5x2 + 3x – 2x?

A) 5x3 + 3x3 + 2 + c 3 2 x

B)4x3 + 3x2 – 22 + c 3 2 x

C)5x3 + 3x2 2 ln| x|+ c 3 2

D)10x + 3 + x22 + c

E)5x2 + 3x – 2x + c

5.f (x) = 3x2 + 2x + 4 and f(1) = 3 are given. What is f(3)?

A) 7	B) 6	C)	1	D) 27	E) 45
			3

6. What is the primitive of the function f(x) = 3 cos x – 4 sin x?

A) 3 sin x + 4 cos x + c

B) 3 sin x – 4 cos x + c

C) –3 sin x – 4 cos x + c

D) –3 sin x + 4 cos x + c

E)	3cos2 x	–2 sin 2x + c
	2

7. (cot2 x+1) dx=?
A) –cot x + c			B) cot x + c
C) sin x + c			D) tan x + c
E)	cot3	x	+ x+ c
	3

8. arccos x dx=?
A) x arccos x + x + c
B) x arccos x +	1 x2	+ c
C) arccosx + x arccos x + c
D) x arccos x –	1 x2	+ c
E) x arccos x +	1+ x2	+ c

Chapter Review Test 1A

77

9. (cos2x 3) dx= ?

A)	sin2x	+ c		B)	sin2x	– 3x+ c
	2				2
C) 2 sin x + c				D) 2 cos x + c
		E)	cos2x	+ x+
			2

10. 9 sin9 x cos x dx= ?

A)	sin10 x	+ c	B)		cos10 x	+ c
	10				10
C)	9sin10 x + c		D)		9cos10 x + c
	10				10
		E)	9sin10 x cos2 x		+ c
			20

11. cos4x cos2x dx= ?

A)	cos4x	+ cos2x		+ c	B)	sin4x sin2x		+ c
	4	2				8
C)	cos6x	+ cos2x		+ c	D)	sin6x	+ sin2x		+ c
	12	4				12	4
		E)	cos x + sin x + c
			2		3

12. cos2 x dx = ?

	A)	sin2 x	+ c		B)	x	+	sin2x	+ c
		2				2		4
	C)	cos3 x		+ c	D) cos x + c
		3

13.f (w)=

A)x2 w

2

C) xw2 2

E) x + cos2x + c 2 4

(xw w) dw is given. What is f(w)?

wx+c B) xw2 2 wx+c

		w2	+ c	D) x2 w2 + w2 + c
		2

E) xw w2 +c

2

14. f (x)= (x2 x 2) dx and f(1) = 2 are given.

	What is f(2)?
	A)	23	B)	17	C)	9	D) 21	E) 13
		6		3		2

78	Algebra 11