36 Find the area by integration of a circle of radius r.
37. Determine the area bounded by the y = 8-x, the x-axis & the y-axis.
A. 14
C. 16
B. 10
D. 12
barrel has the shape of an ellipsoid of revolution with equal pieces cut off from the ends If the barrel
is 12 units long with circular ends of radius 6,4 units and midsection of radius 8 units find the volume
of the barrel. Ans 2, 123.21 cu: units
39. The area of the region bounded by 25(x-3)²- 225 -9y² is revolved about the yaxis. Find the
volume of the solid generated. (Use the Theorem of Pappus).
Ans. 888.26 cu. units
40. Find the area bounded by the parabola x²= 2-y and the line x + y 0. Ans. 4.5 sq. units
41. Find the area formed by the boundaries y 1, x =I and y-x
42. Find the area bounded by parabolas x² =-y+9 and x² = &y
43. Find the volume of the torus formed by revolving the area bounded by (r-2) +y²= 4 about the
line y = 4. Use the theorem of Pappus (2nd Proposition). Ans. 315.83 củ units
44. The area bounded by the parabola x² = 4y and y 4 is revolved about the y-axis. Find the volume of
the paraboloid of revolution generated.
45. The area bounded by the curve y 2*2 and the lines y = 0 and x = 5 is revolved about the y-axIs
Determine the volume generated.
A. 1636 cubic unit
C. 2837 cubic unit
B. 1964 cubic unit
D. 2356 cubic unit
46. Find the volume of the solid of revolution
formed by rotating the region bounded by the parabola y x and the line y = 0 and x 2 about the x-axis.
A. 32/5 pi cubic units
C. 36/5 pi cubic units
B. 34/5 pi cubic units
D. 25/3 pi cubic units
47. A tank full of water is 12 meters long, 4 meters wide and 6 meters deep. How much is the surface of
the water lowered when one third of the work necessary to empty the tank through its top has been
done?: Ans. 3 46 m
48. The area bounded by the parabola y -2(x +2) and y-axis is revolved about the line x = 4. Find the
volume generated.
49, Evaluate the integral of xy with respect to y and then to x, for the limits from x=0 to x = l, and the
limits from y = 1 toy=2.
A. 79
C. 3/4
B. I2
D. I/6
50. Evaluate the integral of sin 8 with respect tó r and then to 8, for the limits from r =0 to r = cos .
and from8 = 0 to 0= n.
A. 1/4
C. 13
B. 1/2
D. 1/6
SI Evaluate the triple integral of xy z d z dydxfor the following limits: z from O to (2-x), y from 0 to
(l-x), and x from 0 to l
A. 85/30
C. 89/30
B. 87/30
D. 81/30
52 A trapezoid area has the following vertices on
the x -y plane: A(6.0, 1.5), B(10.0, 2.50), C(LD.0, -25)
and D(60, -11 5) With all coordinate in cm.
If this area is rotated about the y-axis, determine the
generated volume, in cu cm.
A. 746
C. 821
D. S78
B. 903
$3. Find the volume generated by revolving a rectangle of sides a and b about the side & (Sept 03#20)
A. pix a x (bsquared)
C. pi x (asquared) x b
B. pix a x (bsquared)y2
D pi x (asquared) x b2
54. Revolve the area bounded by a pentagon with vertices (1, 0),(2, 2), (0, 4), (-2, 2), (1, 0) about the xaxis and find the volume generated. (April 05#30)
C.104 pi/3.
A. 36 pi
ot hi 5
D. 38 pi
B. 98 pi/5
55. Given the three vertices of a triangle, (1, 0), (9, 2) and (3, 6). Find the coordinates of intersection of
the
medians.
Flj.
C.(13/3, 8/3)aF ;oie i es
ikt A.(13, 3)
D. (13/2, 4) *iu w'(Sept: 03#40),
ilz
Ersb B, (12/4, 2)
56. The points A(0, 0), B(6, 0) and C(4, 4) are vertices of triangle. Which of the following is an equation
of one medians? (April 05#38)
C. 2x-Sy.=0:.e iü u
A. 2x:-5y = 4
B. 2x + 5y0
D. 2x + Sy=4.ot el ia
57. The area bounded by the hyperbola xy= a', the axis, and the line x 2a revolves about the xaxis. Find the volume generated.
58. A water tank is a horizóntal circular cylinder 10 feet long and ft in diameter. If the water inside is 7.5
feet deep determine the volume of water contained.
A. 663.44.cu ft
C,631.85 cu ft
D. 568.67, cu ft,,
B. 600.26 cu ft
59, A cylinder is generated by a linè moving always parallel-to the line x + z=a, y=0, and following the
curve y² t az = a',x=0. Find the volume in the first octant.inside the cylinder:tat
61. The region bounded by the curve y =x* the x-axis, and 0 the lines x=l and'x= 2 is revolved about
the x-axis. Find the volume of solid of revolution generated.
Ans. 31/5
62. Find the volume of the torus generated by revolving the,circle x + y²= 4about the line x=3.
63. The region bounded by the curve y = x* and the linesy = land x =2 is resolved about the line y =-3.
Find the volume of the solid generated by taking the rectangular elements of area
A) parallel to the axis of revolution
B) perpendicular to the axis of revolution.
64. A solid is constructed with circular base of radius 1 such that every plane section perpendicular to a
certain diameter of the base is isosceles triangle whose altitude is 2. Find the volume of the conoid.
65. October 1985. A hemispherical open tank used for concentration of sulfuric acid (specify gravity
1.25) is filled up to the rim. If the tank has a radius of 3m, detemined the work done in kilojoules after
pumping the tank of all acid to another rank situated on a flatform 3m above the rim of thelower tank.
Ans. 2, 860.39 K.J
66. Find the volume of the solid generated by rotating the curve 4x* +9y' = 36 about the line 3x + 4y 20
67. Find the integration, the Circumference of a circle of radius.
68. Find the length of one branch are the curve 9y = 4x³ from x =0 to x =3,
69. Find the perimeter of the curve x²/3 + y²/3:= a/3.
70. Find the length of the catenary y = a cosh x/a from 0 to x =x,..:
71. Find the surface area of a sphere ofradius a
72. Find the surface area generated by revolvıing the curve y = ax about the x-axis from x =0 to x =a.
73. Find the centroid of the given system. Equal masses at (3, 0), (1, 2), (4, 1), and (-3,3),
74. Find the centroid of the triangle having base equal to b and altitude h.
75. Two posts of equal radius-6 meters and 4 meters tall, stand upright meters apart Find the centroid.
76. The legs of a, table arc 11 meters long; the legs weigh 2 kgs each, the top 5 kgs. Find the c
centroid.
77. From a square of side 2a a circle of radius b, is stamped out. If the circle is tängent to2 sides of the
square, find the centroid.
78. A sphere of radius 3 centimeters rests on s cylinder of radius 4 centimeters and height 8 centimeters.
Find the height of the centroid.
9E
79. Find the centroid of a semi-circular area.
3
A.
4t
4R
3R
3R
B.
4R
D.
41
31t
80. Find the centroid of the area bounded by the curye x= 4x, the x-áxis and the line x=4.
81.Find the centroid of the area in the first quadrant bounded by the parabola y = b'x/a and the line x=
a.
82. Find the centroid the area bounded by the parabolas x² = 2y and x22x+ 2y = 5.
83. Find the centroid of a henisphere of radius r. Ans. č=3/8r
84. Find the centroid of a circular cone. Ans. h from base
85. Find the centroids of the first quadrant are of the circle x+y=r,
86. Find the volume of a wedge out form a circular cone by two planes through-the axis.
87. A tank with its axis horizontal is full of oil (specific gravity 0.80). The ends of the tank are elliptical
in shape with the major and minor axis 4 meters and3 meters respectively. Find the total pressure on
each end.
88. A parabolic plate lowered, vertex downward, until the latus rectum lies in the surface of a liquid.
Find
the force on one side of the plate, if the latus rectum is one mneter long.
89, An elliptical plane, major axis 6 feet minor axis 4 feetis submerged vertically with major axis
horizontal
and its center is 2 feet below the water surface. Find the force on one side of the sübmerged portion of
the plate.
90. An elliptiçal area is submerged so that its major axis is parailel to the surface of the liquid and its
minor
axis vertical. If its uppermost part is c meters below the surface, find the total pressure exérted on the
ellipse. Equation of ellipse: *²/a' + y'/b=1
91. The base of a solid is the circle x +y16. If every plane section perpendicular to a fixed diameter
of the circle is a square, find the volume of the solid. Ans. 341.33 cu. units
INTEGRAL CALCULUS (WORK PROBLEMS) FLUID PRESSURES
1. A spring extends 0.6 inches (x) per 10 lbs weight (y) applied. Supposing when it is stretched 16", we
stretch it some more till it is 2.1" long. Find the work done in stretching the spring from
(a) l.6 to 2.1 in.
Ans. 0,0555 in lb2
b) its,non-stretched length to L6:
Ans. 0.0768 in lbs.
2. Within certain limits, the force required to the stretch a spring is proportional to the stretch, the
constant
of proportionality being called the modulus of the spring. Ifa given spring of normal length 10 in
requires a force of 50 lb to stretch it l/2 inches, calculate the work required to stretch it from
A) 11 to 15 in
B) 12 in tổ l6 in
C)10 in to l6 in
3. A vertical cistern of 5 meters diameter and a depth of6 meters is filled with water weighing 9.81 x 10N
per cubic meter. Calculate the work necessary to pump the water to the top of the cistern,
A) if the cistern is full of water
B) if the cistern is half full of water:
4. A vertical cylindrical tank is 2 meters in diameter and 3 meter high is filled water.
A, find the amount of work done in pumping all the water out at the top of the tank, if the tank is half
full of water.
B. find the amount of work done in pumping half the water out at the top of the tank, if the tank is full
of water.
5. A vessel in the form of a right circular cone id filled with water. Ifh is its height and r radius of the base,
what time will it require to empty itself through an orifice of area a
a at thé vertex?
20. A horizontal cylindrical tank of 8 feet diameter is filled with oil.weighing 60 lb per cubic foot.
Calculate:
22. The vertical end of a vat is a segment of a parabola,(with vetex at the bottom) 8 feet across the top
and
9. A hemispherical tank of diameter of
6. A conical reservoir 4 meters deep is filled with a liquid weighing 10 x10*N/m. The top of the reservoir
7. A conical cistern in 6 meters across the top and 4 meters deep. If the surtace of the water is 2 meters
19. A leaking sack of flour is to be raised to a height of4 meters. The loss in Newtons due to leakage is
12. A water tank is in the form of a hemisphere, 8 meters in diameter, surmounted bý a cylinder of the
same
13. A water tank is cylindrical in shape with a hemispherical bottom: The cylindrical portion has a 4
meter
15. March 1977 (CE May 1979). A Cistern in the form of an inverted right circular cone is 20m deep and
17. A canal 100 feèt long and having a semi-circular cross-section5 feet in radius id full of mud wate-(w
16. A tank has the shape of a paraboloid of revolution. The radius of the circular top is 2 meters and its
18 A well is 30 meters deep. A bucket, weighing 12N, has a volume of 0.01 m. The bucket is filled with
23. Each end of a horizontal oil tank is an ellipse of which the horizontal axis is 4 meters long and the
8.
10. A tank of hemispherical shape 2,meters in diameter is full of water. How much work is done in
emptying
21. The vertical end of water trough is an 1sosceles triangle 2 meters across the topand 2 meters deep.
14. Find the work done in pumping out a semi-elliptical reservoir full of water. The top ís a circle of 2
meter
24. A rectangular gate 2 meters wide and 3 meters high is submerged vertically with its upper edge at
the
Calculate the network done in pumping out the water.filling a hemisphericał reservoir 3 meters deep.
diameter and 2 meters altitude.The tank is filled through its bottom, which is 13 meters above the
necessom.
proportional to the square root of the distance from the original position. if the initial weight of the sack
the top.
Calculate the pressure on the end when the trough is full of water:.
is 240N and the total loss is 50N, fnd the work done.
the tank if all the water
water is leaking out at a constant rater of l*10% cu, meter per second.
Neglecting the weight of the rope, find the work done in raising the bucket if it is discovered that the
diameter and 3 meters high. Find the work done in pumping it'out whenit is filled within1 meter of
below the top, find the work done in pumping the water to the top of the cistern.
reservO1r.
weighing 94x10³ N De
vertical axis 2 meters long. Calculate the pressure, on one end.when the tank is half full of oil
foot.
intake to the pump.
diameter and the depth is 2 meters.
is a circle 2 meters in diameter. Calculatethe
surface of the water. Find the total prèssure and the location of the center of pressure.
water at the bottom of the well and is then raised at a constant, rate of 2 meters per second to the top.
at a point 2 meters above the top of the tank.
height is 4 meters. If the tank is full of water, find the work required to empty the water from the tank
Ans. 68166750N-m
12m diameter at the top. If the water is l6m deep in the cistern, find the work done in pumping out the
70 lb/ft). The water is to be pumped to a point 2 feet above the canal. What hp, capacity pump must
be used to clear the canal in 1 hour? Assume pump, efficiency of 90%.
16 feet deep. Calculate the pressure at this end when the vat is full of a liquid weighing 70 b pér cubic
water. The water is raised to a point of discharge 1Om.above the top of the cistern. Neglect all frictions.
A) the pressure on one end if the tank is half full
B) if the pump used in filling. the tank develops 2 kW, how long willit take to fill the tank?
B) the pressüre on one end if the tank is full
A) compute the work done by the pump in filling the tank
to pump the oil
per cubic meter.-;
must be pumped up.over edge of the tank?
to the top of the
3 meters is full of oilweighing
tank.
work necessary to pump the
9.2 x10*N /m$ Calculate
liquid to the top of the
the work