COCOON TEST MATH 8 – Integral Calculus 2
-------------------------------------------------------------------------LARVA TEST
Test Your Retentivity
4. Find the length of the curve given its
parametric equations x=t3-3t and y=3t2 form t=0
to t=1.
A. 2
B. 3
C. 4
D. 5
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11. Find the volume of the solid formed by
rotating the curve 4x^2 + 9y^2 = 36 about the
line 4x + 3y – 20=0.
A. 473.74
B. 130.54
C. 210.64
D. 320.65
12. The region bounded by the curve y = x^2, x
= 2, and y = 0 is rotated about the line x = 2.
Find the volume of the solid obtained.
A. 5pi/3
B. 16pi/3
C. 8pi/3
D. 10pi/3
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5. Find the centroid of the region bounded by y
= x2, y = 0, and x = 1.
A. (1/4, 2/3)
C. (3/4, 3/10)
B. (2/3, 5/4)
D. (3/5, 5/10)
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3. Find the length of the curve having a
parametric equations of x= a cos^3 θ y=a sin^2
θ from θ=0 to θ=2π.
A. 5a
B. 6a
C. 7a
D. 8a
A. IPP = 625,000 cm4 , k = 32.29 cm
B. IPP = 635,000 cm4 , k = 32.53 cm
C. IPP = 645,000 cm4 , k = 32.79 cm
D. IPP = 655,000 cm4 , k = 33.04 cm
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2. Find the length of arc of a circle x2+y2=25
from x=4 to x=2.
A. 2.58
B. 5.28
C. 8.65
D. 6.85
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1. Find the length of the curve r=a (1-cos θ)
from θ=0 to θ=π and also the total length of
curve.
A. 4a; 8a
C. 3a; 6a
B. 2a; 4a
D. 5a; 10a
6. Find the centroid of the area bounded by x^2
= 4 – y, the x-axis and the y-axis on the first
quadrant.
A. (3/4, 5/8)
B. (3/8, 5/2)
C. (3/8, 2/5)
D. (3/4, 8/5)
7. Find the moment of inertia, with respect to
x-axis of the area bounded by the parabola
y^2=4x and the line x=1.
A. 4.12
B. 2.13
C. 3.16
D. 5.18
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8. Determine the second moment of area and the
radius of gyration about the axis AA for the
rectangle shown below.
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A. IAA = 2304 cm4 , k = 6.93cm
B. IAA = 256 cm4 , k = 2.31cm
C. IAA = 64 cm4 , k = 1.15cm
D. None of the above
9. Determine the second moment of area and the
radius of gyration about the axis C for the
rectangle shown below.
A. IC = 2304 cm4 , k = 6.93cm
B. IC = 256 cm4 , k = 2.31cm
C. IC = 64 cm4 , k = 1.15cm
D. None of the above
10. Find the second moment of area and the
radius of gyration about axis PP for the
rectangle shown
13. Find the volume of the solid revolution
obtained by revolving the region bounded by
y=x-x2 and the x axis about the x axis.
A.π/15
C. π/30
B.π/45
D. π/60
14. Find the volume obtained if the region
bounded by y = x2 and y = 2x is rotated about
the x axis.
A.34π/15
C. 54π/5
B.64π/15
D. 14π/5
15. Compute the volume of the solid obtained by
rotating the region bounded by y = x2, y = 8 –
x2, and the y axis about the x axis.
A.156π/3
C. 254π/3
B.256π/3
D. 356π/3
16. A hole of radius 2 is drilled through the
axis of a sphere of radius 3. Compute the
volume of the remaining solid.
A.46.832
C. 38.234
B.35.235
D. 50.234
17. Given the area in the first quadrant
bounded by x2=8y, the line x=4 and the x-axis.
What is the volume generated by revolving this
area about the y-axis?
A.40. 525 cu. units
B.25.134 cu. units
C. 50.265 cu. units
D. 38.625 cu. Units
18. The area on the first and second quadrant
of the circle x2+y2=36 is revolved about the
line y=6. What is the volume generated?
A.1235.80 cu. units
B.1245.80 cu. units
C. 1225.80 cu. units D.1325.80 cu. Units
19. The area bounded by a curve y2=12x and the
line x=3 is revolved about the line x=3. What
is the volume generated?
A. 180
B. 186
C. 184
D. 181
20. The region bounded by y = x2, x = 2 and y =
0 is rotated about the line x = 2. Find the
volume of the solid obtained.
A. 5π/3
B. 16π/3
C. 8π/3
D. 10π/3
COCOON TEST MATH 8 – Integral Calculus 2
-------------------------------------------------------------------------21. Find the surface area generated by rotating
the first quadrant portion of the curve x2=16 8y about the y-axis.
A.64.89
B.61.27
C. 76.13
D. 74.28
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22. The parabolic reflector of an automobile
headlight is 12 cm in diameter and 4 cm in
depth. What is the surface area in cm2 ?
A. 153.94
B. 135.97
C. 127.82
D. 156.
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25. An object experiences rectilinear
acceleration a(t)=10-2t . How far does it
travel in 6 seconds if its initial velocity is
10m/s.
A.
254 m
C.168 m
B.
287 m
D. 133 m
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24. A spring with a natural length of 10 cm, is
stretched by 1/2 cm. by a 12 Newton force. Find
the work done in stretching the spring from 10
cm. to 18 cm. Express your answer in joules.
A.6.68 Joules
B.7.68 Joules
C. 14.68 Joules
D. 10.68 Joules
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23. Determine the moment of inertia with
respect to x-axis of the region in the first
quadrant which is bounded by the curve y2=4x,
the line y=2 and the y-axis.
A.1.6
C. 1.9
B.1.3
D. 1.5
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COCOON TEST MATH 8 – Integral Calculus 2
-------------------------------------------------------------------------PUPA TEST
Test Your Board Exam Readiness
circumference of the smaller circle is given by
𝑥𝑥 = 5 cos(𝑡𝑡) − cos(5𝑡𝑡) and 𝑦𝑦 = 5𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝑡𝑡) − sin(5𝑡𝑡). Find the
distance traveled by the point in one complete
trip about the larger circle.
A. 20
B. 30
C. 40
D. 50
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16. Let C be the arc of the circle 𝑥𝑥 2 + 𝑦𝑦 2 = 9
3√3
𝜋𝜋
2
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from (3,0) to (3/2, 2 ). Find area of the
surface formed by revolving about the axis.
A. 7pi
B. 8pi
C. 9pi
D. 10pi
17. Find the area of one petal of the rose
curve given by r=3cos(3θ).
A. 3pi/4
B. 3pi/5
C. 3pi/6
D. 3pi/7
18. Find the length of the arc from 𝜃𝜃 = 0to 𝜃𝜃 = 2𝜋𝜋
for the cardioid 𝑟𝑟 = 𝑓𝑓(𝜃𝜃) = 2 − 2cos(𝜃𝜃)
A. 8
B. 16
C. 32
D. 64
19. Find the area of the surface formed by
revolving the circle 𝑟𝑟 = cos(𝜃𝜃) about the line 𝜃𝜃 =
A. 2pi
B. pi^2
C. 2pi^2
D. pi
20. Determining a Conic from Its Equation 𝑟𝑟 =
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1. A force of 40 N is required to hold a spring
that has been stretched from its natural length
of 10 cm to a length of 15 cm. How much work is
done in stretching the spring from 15 cm to 18
cm?
A. 3.16 J
B. 2.13 J
C. 1.56 J
D. 4.22 J
2. A 40 metre long cable that weighs 5 kg/metre
is hanging from the roof of a very tall
building. How much work is required to lift it
all to the roof level?
A. 39,200 J
B. 49,200 J
C. 59,200 J
D. 69,200 J
3. A conical water tank of height 8m and
diameter 6m at the base must have all of its
fluid contents pumped to the top of the tank.
If it is full to a depth of 4m, how much work
will this require?
A. 162.22kJ
B. 262.22kJ
C. 462.22kJ
D. 662.22kJ
4. How much work is required to pump all the
water from a right circular cylindrical tank
that is 4 feet in diameter and 8 feet tall, if
it is emptied at a point 2 feet above the top
of the tank?
A. 49421π ft-lb
B. 11981π ft-lb
C. 54481π ft-lb
D. 13171π ft-lb
5. Let L be the length of the curve with
parametric equations x = ln t and y = sin t
with limits from 1 to π. Determine L.
A. 1.3477
B. 1.7734
C. 1.3734
D. 1.7374
6. Find the length of the arc of x^2 + y^2 = 64
from x=-1 to x=-3, in the second quadrant.
A. 3.15
B. 3.22
C. 2.07
D. 2.16
7. Locate the centroid of the area bounded by
the parabola, the line y=4 and the y-axis.
A. 6/5, 3
B. 2/5, 3
C. 3/5, 3
D. 4/5, 3
8. Find the distance of the centroid from the
y-axis of the area bounded by the curve
x^2=16y, the line x=12 and the x-axis.
A. 8
B. 9
C. 4
D. 3
9. Find the center of mass of a system of point
masses m1=6,m2=3,m3=2 and m4=9, located at
points (3,-2), (0,0), (-5,3) and (4,2)
respectively.
A. (11/3,3/5)
B. (11/4,3/5)
C. (11/5,3/5)
D. (11/6,3/5)
10. Find the center of mass of the lamina of
uniform density bounded by the graph of f(x)=4x^2 and the axis.
A. (0,9/5)
B. (0,8/5)
C. (0,7/5)
D. (0,6/5)
11. Find the centroid of the region bounded by
the graphs of f(x)=4-x^2 and g(x)=x+2
A. (-1/3,12/5)
B. (-1/3,13/5)
C. (-1/2,12/5)
D. (-1/2,13/5)
12. Find the volume of the torus which was
formed by revolving the circular region bounded
by (𝑥𝑥 − 2)2 + 𝑦𝑦 2 = 1 about the y-axis.
A. 39.5
B. 49.5
C. 59.5
D. 69.5
13. Find the length of the parabolic x^2=4py
arc intercepted by the latus rectum.
A. 3.59p
B. 4.59p
C. 5.59p
D. 6.59p
14. What is the circumference of the ellipse
𝑥𝑥 2
𝑦𝑦2
+ 16 = 1
A. 22.44 units
B. 24.44 units
C. 26.44 units
D. 28.44 units
15. A circle of radius 1 rolls around the
circumference of a larger circle of radius 4,
The epicycloid traced by a point on the
25
15
3−2cos(𝜃𝜃)
A. Circle
B. Parabola
C. Ellipse
D. Hyperbola
21. Determining a Conic from Its Equation 𝑟𝑟 =
32
3+5𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃)
A. Circle
B. Parabola
C. Ellipse
D. Hyperbola
22. Find the moments of inertia of the triangle
bounded by 3x+ 4y = 24, x = 0, and y = 0, with
respect to the x-axis.
A. 124
B. 134
C. 144
D. 154
23. A water tank in the shape of a
hemispherical bowl of radius 4 m is filled with
water to a depth of 2 m. How much work is
required to pump all the water over the top of
the tank?
A. 1169.48kJ
B. 1149.48kJ
C. 1129.48kJ
D. 1109.48kJ
24. A conical tank, 10 meters deep and 8 meters
across at the top, is filled with water to a
depth of 5 meters. The tank is emptied by
pumping the water over the top edge. How much
work is done in the process?
A. 1353kJ
B. 1363kJ
C. 1273kJ
D. 1283kJ
1
25. The region bounded by 𝑦𝑦 = 𝑥𝑥 , 𝑦𝑦 = 0, 𝑥𝑥 = 0 find
the centroid.
1
1
A. �ln(2) , 4 ln(2)�
C. �
1
,
1
ln(4) 4 ln(4)
�
1
1
B. �ln(3) , 4 ln(3)�
D. �
1
,
1
ln(5) 4 ln(5)
�
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