Mathematics, Surveying, and Transportation Engineering

INTEGRAL CALCULUS

INTEGRAL CALCULUS b. 18 cm d. 45 cm

Situation: The motion of a particle is defined by the parametric

INDEFINITE INTEGRALS
equations ax = 0.8t, ay = 2 – 0.3t, and az = 5, where a is in m/s2
1. Evaluate:
and t in seconds. The particle is at rest and is placed at the origin
𝑥 3 + 3𝑥 2 + 𝑥
∫ 𝑑𝑥 at t = 0.
𝑥−3
8. What is the acceleration of the particle after 10
a. (1/3)x3 + 3x2 + 19x + 57ln(x – 3) + C
seconds?
b. (1/3)x3 - 3x2 - 19x - 57ln(x – 3) + C
a. 10.71 m/s2 c. 9.49 m/s2
c. (1/3)x3 + 3x2 - 19x - 57ln(x – 3) + C 2
b. 8.46 m/s d. 11.16 m/s2
d. (1/3)x3 - 3x2 + 19x - 57ln(x – 3) + C
9. What is the velocity of the particle after 10 seconds?
2. Evaluate:
a. 64.23 m/s c. 57.09 m/s
∫ 𝑥𝑒 2𝑥 𝑑𝑥 b. 70.25 m/s d. 67.13 m/s
a. (1/2)xe2x + (1/4)e2x + C 10. What is the displacement of the particle after 10
b. (-1/2)xe2x + (1/4)e2x + C seconds?
c. (-1/2)xe2x - (1/4)e2x + C a. 281.88 m c. 295.08 m
d. (1/2)xe2x - (1/4)e2x + C b. 287.71 m d. 291.61 m
11. What is the distance traveled by the particle after 10
DEFINITE INTEGRALS seconds?
3. Evaluate: a. 281.88 m c. 295.08 m
1 b. 287.71 m d. 291.61 m
∫ (5x+3)2 dx
0
PLANE AREAS
a. 97/3 c. 97/6 12. Find the area of the region bounded by the curves 𝑦 =
12𝑥
b. – 97/3 d. – 97/6 , the x – axis, x = 1, and x = 4.
𝑥 2 +4
4. Evaluate: a. 6 ln(4) c. 3 ln(5)
𝜋/2
b. 4 ln(6) d. 5 ln(3)
∫ cos 7 𝑥 sin6 𝑥 𝑑𝑥 13. Determine the area bounded by the curve x2 = 5y and x
0 = 3y.
a. 8.89 x 10-4 c. 7.22 x 10-3 a. 0.145 c. 0.451
b. 5.33 x 10-3 d. 6.45 x 10-4 b. 0.154 d. 0.415
14. What is the area within the curve r = 2cos 3θ?
MULTIPLE INTEGRALS a. 2 sq. units c. 6 sq. units
5. Evaluate: b. 4 sq. units d. 8 sq. units
3 2𝑦
15. Find the area enclosed inside the polar curve r2 = 10
∫ ∫ (𝑥 2 + 𝑦 2 ) 𝑑𝑥 𝑑𝑦 cos(2θ).
1 0 a. 5 c. 15
a. 260/3 c. 280/3 b. 10 d. 20
b. 270/3 d. 290/3
VOLUME OF SOLID OF REVOLUTION
DISPLACEMENT, VELOCITY AND ACCELERATION 16. Consider the area bounded by the curve x2 = 5y and x
6. The acceleration is given by a = 2 + 12t, where t is in = 3y, what is the volume generated when the area is
minute and a in m/min2. The velocity of the particle is rotated about the x-axis?
11 m/min after 1 minute. Find the velocity after 2 a. 0.251 c. 0.215
minutes. b. 0.154 d. 0.326
a. 31 m/min c. 35 m/min 17. Consider the area bounded by the curve x2 = 5y and x
b. 33 m/min d. 37 m/min = 3y, what is the volume generated when the area is
7. The position function of an object that moves on a rotated about the y-axis?
coordinate line is S(t) = t² - 6t where S is measured in a. 0.880 c. 0.990
b. 0.909 d. 0.808
centimeters and t in seconds. Find the distance travelled 18. Find the volume generated by revolving the triangle
in the time interval (0,9). whose vertices are (2, 2), (4, 8), and (6, 2) about the
a. 36 cm c. 27 cm line 3x – 4y = 12.

John Rey M. Pacturanan, CE, MP Page 1 of 2

 Mathematics, Surveying, and Transportation Engineering
INTEGRAL CALCULUS

a. 241.27 c. 217.47 b. 44.83 kJ d. 42.84 kJ

b. 214.27 d. 242.17
HYDROSTATIC FORCES
CENTROIDS 28. A rectangular plate is 4 feet long and 2 feet wide. It is
19. Determine the coordinates of the centroid of the area submerged vertically in water with the upper 4-ft edge
bounded by the curve x2 = 5y and x = 3y. parallel to the water surface and is 3 ft below the
a. (5/6, 2/9) c. (1/6, 4/9) surface. Find the magnitude of the resultant force
b. (2/9, 5/6) d. (4/9, 1/6) against one side of the plate. Assume w is the unit
weight of water.
MOMENT OF INERTIA a. 24w c. 48w
20. Determine the moment of inertia with respect to the x- b. 32w d. 64w
axis of the area bounded by the curve x2 = 5y and x = 29. Find the force on one face of a right triangle with sides
3y. 4 m and altitude 3 m. The altitude is submerged
a. 0.0603 c. 0.0306 vertically with the 4-m side in the surface.
b. 0.0360 d. 0.0630 a. 58.86 kN c. 56.88 kN
21. Determine the moment of inertia with respect to the y- b. 58.68 kN d. 56.68 kN
axis of the area bounded by the curve x2 = 5y and x =
3y.
a. 0.192 c. 0.291
b. 0.129 d. 0.219

LENGTH OF ARC
22. What is the length of the curve y = x3/3 + 1/4x from x
= 1 to x = 2?
a. 2.5483 c. 2.4538
b. 2.4583 d. 2.5438
23. Find the length of one arc of the curve whose
parametric equations are x = 6θ – 6sinθ and y = 6 –
6cosθ.
a. 24 units c. 40 units
b. 32 units d. 48 units
24. What is the perimeter of r = 3(1 + cosθ)?
a. 12 units c. 24 units
b. 18 units d. 30 units

WORK
25. A cylindrical tank with a base radius of 5 ft. and a
height of 20 ft. is filled with water to a height of 5 ft.
Find the work done in pumping all the water out of the
top of the tank.
a. 428.8 kip-ft c. 482.8 kip-ft
b. 248.8 kip-ft d. 284.8 kip-ft
26. A conical reservoir with top radius of 4’ and a height of
10’ is full of oil (49 pcf). Find the work done in ft-lb
needed to pump all the oil out of the top of the tank.
a. 20255 c. 20525
b. 20522 d. 20552
27. An open top hemispherical tank having a radius of 1.2
m is full of water. It is to be drained by a vertical pipe
whose exit is 0.80 m above the top of the tank. How
much work is done in emptying the full content of the
tank?
a. 42.48 kJ c. 44.38 kJ

John Rey M. Pacturanan, CE, MP Page 2 of 2