This is the Multiple Choice Questions Part 2 of the Series in Integral Calculus topic in Engineering Mathematics. In Preparation for the ECE Board Exam make sure to expose yourself and familiarize in each and every questions compiled here taken from various sources including but not limited to past Board Examination Questions in Engineering Mathematics, Mathematics Books, Journals and other Mathematics References.

### Multiple Choice Questions Topic Outline

- MCQs in Basic Integrals | MCQs in Integrals of Exponential Functions | MCQs in Integrals of Logarithmic Functions | MCQs in Integrals of Trigonometric Functions | MCQs in Integrals in Inverse Trigonometric Functions | MCQs in Integrals of Hyperbolic Functions | MCQs in Integrals of Trigonometric Substitution | MCQs in Integration by parts | MCQs in Integral involving Plane Areas | MCQs in Integral involving Centroid | MCQs in Integral involving Length of Arc | MCQs in Integral involving Propositions of Pappus | MCQs in Integral involving Work | MCQs in Integral involving Moment of Inertia

### Online Questions and Answers in Integral Calculus Series

Following is the list of multiple choice questions in this brand new series:

**Integral Calculus MCQs**

**MCQs from Number 1 – 50**Answer key:

**PART I**

**MCQs from Number 51 – 100**Answer key:

**PART II**

### Continue Practice Exam Test Questions Part I of the Series

**Choose the letter of the best answer in each questions.**

51.) Find the total length of the curve r = 4(1 – SinÎ¸) from Î¸ = 90Âº to Î¸ = 270Âº and also the total perimeter of the curve.

- a. 12, 24
- b. 15, 30
- c. 16, 32
- d. 18, 36

52. Find the length of the curve r = 4Sin Î¸ from Î¸ = 0Âº to Î¸ = 90Âº and also the total length of curve.

- a. Ï€ ; 2Ï€
- b. 2Ï€ ; 4Ï€
- c. 3Ï€ ; 6Ï€
- d. 4Ï€ ; 8Ï€

53. Find the length of the curve r = a (1 – CosÎ¸) from Î¸ = 0Âº to Î¸ = Ï€ and also the total length of the curve.

- a. 2a ; 4a
- b. 3a ; 6a
- c. 4a ; 8a
- d. 5a ; 9a

54. Find the total length of the curve r = a CosÎ¸.

- a. Ï€a
- b. 2Ï€a
- c. 1.5Ï€av
- d. 0.67Ï€a

55. Find the length of the curve having a parametric equations of x = a Cos^{3}Î¸, y = a Sin^{2}Î¸ from Î¸ = 0Âº to Î¸ = 2Ï€.

- a. 5a
- b. 6a
- c. 7a
- d. 8a

56. Find the centroid of the area bounded by the curve y = 4 – x^{2}, the line x = 1 and the coordinate axes.

- a. (0.24, 1.57)
- b. (1.22, 0.46)
- c. (0.48, 1.85)
- d. (2.16, 0.53)

57. Find the centroid of the area under y = 4 – x^{2} in the first quadrant.

- a. (0.75, 1.6)
- b. (1.6, 0.95)
- c. (0.74, 1.97)
- d. (3.16, 2.53)

58. Find the centroid of the area in first quadrant bounded by the curve y^{2 }= 4ax and the latus rectum.

- a. (0.6a, 0.75a)
- b. (1.23a, 0.95a)
- c. (0.94a, 2.97a)
- d. (1.16a, 0.53a)

59. A triangular section has coordinates of A(2,2), B(11,2), and C(5,8). Find the coordinates of the centroid of the triangular section.

- a. (7, 4)
- b. (6, 4)
- c. (8, 4)
- d. (9, 4)

60. The following cross section has the following given coordinates. Compute for the centroid of the given cross section. A(2,2), B(5,8), C(7,2), D(2,0), and E(7,0).

- a. (4.6, 3.4)
- b. (4.8, 2.9)
- c. (5.2, 3.8)
- d. (5.3, 4.1)

61. Sections ABCD is a quadrilateral having the given coordinates A(2,3), B(8,9), C(11,3), and D(11,0). Compute for the coordinates of the centroid of the quadrilateral.

- a. (5.32, 3)
- b. (6.23, 4)
- c. (7.33, 4)
- d. (8.21, 3)

62. A cross section consists of a triangle and a semi circle with AC as its diameter. If the coordinates of A(2,6), B(11,9), and C(14,6). Compute for the coordinates of the centroid of the cross section.

- a. (4.6, 3.4)
- b. (4.8, 2.9)
- c. (5.2, 3.8)
- d. (5.3, 4.1)

63. A 5m x 5cm is cut from a corner of 20cm x 30cm cardboard. Find the centroid from the longest side.

- a. 10.99 m
- b. 11.42 m
- c. 10.33 m
- d. 12.42 m

64. Locate the centroid of the area bounded by the parabola y^{2 }= 4x, the line y = 4 and the y-axis.

- a. (0.4, 3)
- b. (0.6, 3)
- c. (1.2, 3)
- d. (1.33, 3)

65. Find the centroid of the area bounded by the curve x^{2 }= –(y – 4), the x-axis and the y-axis on the first quadrant.

- a. (0.25, 1.8)
- b. (1.25, 1.4)
- c. (1.75, 1.2)
- d. (0.75, 1.6)

66. Locate the centroid of the area bounded by the curve y^{2 }= -1.5(x – 6), the x-axis and the y-axis on the first quadrant.

- a. (2.2, 1.38)
- b. (2.4, 1.13)
- c. (2.8, 0.63)
- d. (2.6, 0.88)

67. Locate the centroid of the area bounded by the curve 5y^{2 }= 16x and y^{2 }= 8x – 24 on the first quadrant.

- a. (2.20, 1.51)
- b. (1.50, 0.25)
- c. (2.78, 1.39)
- d. (1.64, 0.26)

68. Locate the centroid of the area bounded by the parabolas x^{2 }= 8y and x^{2 }= 16(y – 2) in the first quadrant.

- a. (3.25, 1.2)
- b. (2.12, 1.6)
- c. (2.67, 2.0)
- d. (2.00, 2.8)

69. Given the area in the first quadrant bounded by x^{2 }= 8y, the line y – 2 = 0 and the y-axis. What is the volume generated when revolved about the line y-2=0?

- a. 53.31 m
^{3} - b. 45.87 m
^{3} - c. 26.81 m
^{3} - d. 33.98 m
^{3}

70. Given the area in the first quadrant bounded by x^{2 }= 8y, the line x = 4 and the x-axis. What is the volume generated by revolving this area about the y-axis?

- a. 78.987 m
^{3} - b. 50.265 m
^{3} - c. 61.253 m
^{3} - d. 82.285 m
^{3}

71. Given the area in the first quadrant bounded by x^{2 }= 8y, the line y – 2 = 0 and the y-axis. What is the volume generated when this area is revolved about the x-axis.

- a. 20.32 m
^{3} - b. 34.45 m
^{3} - c. 40.21 m
^{3} - d. 45.56 m
^{3}

72. Find the volume formed by revolving the hyperbola xy = 6 from x = 2 to x = 4 about the x-axis.

- a. 23.23 m
^{3} - b. 25.53 m
^{3} - c. 28.27 m
^{3} - d. 30.43 m
^{3}

73. The region in the first quadrant under the curve y = Sinh x from x = 0 to x = 1 is revolved about the x-axis. Compute the volume of solid generated.

- a. 1.278 m
^{3} - b. 2.123 m
^{3} - c. 3.156 m
^{3} - d. 1.849 m
^{3}

74. A square hole of side 2 cm is chiseled perpendicular to the side of a cylindrical post of radius 2 cm. If the axis of the hole is going to be along the diameter of the circular section of the post, find the volume cutoff.

- a. 15.3 m
^{3} - b. 23.8 m
^{3} - c. 43.7 m
^{3} - d. 16.4 m
^{3}

75. Find the volume common to the cylinders x^{2 }+ y^{2 }= 9 and y^{2 }+ z^{2 }= 9.

- a. 241m
^{3} - b. 533m
^{3} - c. 424m
^{3} - d. 144m
^{3}

76. Given is the area in the first quadrant bounded by x^{2 }= 8y, the line, the line x = 4 and the x-axis. What is the volume generated by revolving this area about the y-axis.

- a. 50.26m
^{3} - b. 52.26m
^{3} - c. 53.26m
^{3} - d. 51.26m
^{3}

77. The area bounded by the curve y^{2 }= 12x and the line x = 3 is revolved about the line x = 3. What is the volume generated?

- a. 185
- b. 187
- c. 181
- d. 183

78. The area in the second quadrant of the circle x^{2 }+ y^{2 }= 36 is revolved about the line y + 10 = 0. What is the volume generated?

- a. 2128.63
- b. 2228.83
- c. 2233.43
- d. 2208.53

79. The area enclosed by the ellipse 0.11x^{2 }+ 0.25y^{2 }= 1 is revolved about the line x = 3, what is the volume generated?

- a. 370.3
- b. 360.1
- c. 355.3
- d. 365.1

80. Find the volume of the solid formed if we rotate the ellipse 0.11x^{2 }+ 0.25y^{2 }= 1 about the line 4x + 3y = 20.

- a. 40 Ï€
^{ 2}m^{3} - b. 45Ï€
^{2}m^{3} - c. 48 Ï€
^{ 2}m^{3} - d. 53 Ï€
^{ 2}m^{3}

81. The area on the first and second quadrant of the circle x**^{2 }**+ y

**= 36 is revolved about the line x = 6. What is the volume generated?**

^{2 }- a. 2131.83
- b. 2242.46
- c. 2421.36
- d. 2342.38

82. The area on the first quadrant of the circle x**^{2 }**+ y

**= 25 is revolved about the line x = 5. What is the volume generated?**

^{2 }- a. 355.31
- b. 365.44
- c. 368.33
- d. 370.32

83. The area of the second and third quadrant of the circle x**^{2 }**+ y

**= 36 is revolved about the line x = 4. What is the volume generated?**

^{2 }- a. 2320.30
- b. 2545.34
- c. 2327.25
- d. 2520.40

84. The area on the first quadrant of the circle x**^{2 }**+ y

**= 36 is revolved about the line y + 10 = 0. What is the volume generated?**

^{2 }- a. 3924.60
- b. 2229.54
- c. 2593.45
- d. 2696.50

85. The area enclosed by the ellipse 0.0625x**^{2 }**+ 0.1111y

**= 1 on the first and 2**

^{2 }^{nd}quadrant, is revolved about the x-axis. What is the volume generated?

- a. 151.40
- b. 155.39
- c. 156.30
- d. 150.41

86. The area enclosed by the curve 9x**^{2 }**+ 16y

**= 144 on the first quadrant, is revolved about the y-axis. What is the volume generated?**

^{2 }- a. 98.60
- b. 200.98
- c. 100.67
- d. 54.80

87. Find the volume of an ellipsoid having the equation 0.04x^{2 }+ 0.0625y^{2 }+ 0.25z^{2 }= 1.

- a. 167.55
- b. 178.40
- c. 171.30
- d. 210.20

88. Find the volume of a spheroid having equation 0.04x^{2 }+ 0.111y^{2 }+ 0.111z^{2 }= 1.

- a. 178.90
- b. 184.45
- c. 188.50
- d. 213.45

89. The region in the first quadrant which is bounded by the curve y^{2 }= 4x, and the lines x = 4 and y = 0, is revolved about the x-axis. Locate the centroid of the resulting solid revolution.

- a. 2.667
- b. 2.333
- c. 1.111
- d. 1.667

90. The region in the first quadrant, which is bounded by the curve x^{2 }= 4y, the line x = 4, is revolved about the line x = 4. Locate the centroid of the resulting solid revolution.

- a. 0.6
- b. 0.5
- c. 1.0
- d. 0.8

91. The area bounded by the curve x^{3 }= y, the line y = 8 and the y-axis, is to be revolved about the y-axis. Determine the centroid of the volume generated.

- a. 4
- b. 5
- c. 6
- d. 7

92. The area bounded by the curve y = x^{3 }and the x-axis. Determine the centroid of the volume generated.

- a. 2.25
- b. 1.75
- c. 1.25
- d. 0.75

93. Find the moment of inertia of the area bounded by the curve x^{2 }= 4y, the line y = 1 and the y-axis on the first quadrant with respect to x-axis.

- a. 1.2
- b. 3.5
- c. 0.57
- d. 1.14

94. Find the moment of inertia of the area bounded by the curve x^{2 }= 4y, the line y = 1 and the y-axis on the first axis with respect to y axis.

- a. 6.33
- b. 1.07
- c. 0.87
- d. 0.94

95. Find the moment of inertia of the area bounded by the curve x^{2 }= 8y, the line x = 4, and the x-axis on the first quadrant with respect to x-axis.

- a. 1.52
- b. 2.61
- c. 1.98
- d. 2.36

96. Find the moment of inertia of the area bounded by the curve x^{2 }= 8y, the line x = 4, and the x-axis on the first quadrant with respect to y-axis.

- a. 21.8
- b. 25.6
- c. 31.6
- d. 36.4

97.) Find the moment of inertia of the area bounded by the curve y^{2 }= 4x, the line x = 1, and the x-axis on the first quadrant with respect to x-axis.

- a. 1.067
- b. 1.142
- c. 1.861
- d. 1.232

98. Find the moment of inertia of the area bounded by the curve y^{2 }= 4x, the line x = 1, and the x-axis on the first quadrant with respect to y-axis.

- a. 0.436
- b. 0.682
- c. 0.571
- d. 0.716

99. Find the moment of inertia of the area bounded by the curve y^{2 }= 4x, the line y = 2, and the y-axis on the first quadrant with respect to y-axis.

- a. 0.064
- b. 0.076
- c. 0.088
- d. 0.095

100. Find the moment of inertia with respect to x-axis of the area bounded by the parabola y^{2 }= 4x, the line x = 1.

- a. 2.13
- b. 2.35
- c. 2.68
- d. 2.56

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