ENGINEERING CORRELATION COURSE 1
(ENGINEERING MATHEMATICS)
ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#1
∫
Evaluate x 2 x 3 + 3 dx
2 3 3
(x + 3) + C
2
A.
9
1 3 3
(x + 3) + C
2
B.
3
1 3 −1
(x + 3) + C
2
C.
3
x 3 3
(x + 3) + C
2
D.
3
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#2
∫
Evaluate x 2e x dx
A. xe x (x 2 + 2x + 2) + C
B. e x (x 2 − 2x + 2) + C
C. xe x (x 2 + 2) + C
D. e x (x − 1) + C
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#3
∫
Evaluate sec4 x tan6 x dx
1 1
A. tan9 x + tan7 x + C
9 7
1 1
B. tan9 x − tan7 x + C
7 9
1 1
C. tan7 x + tan9 x + C
9 7
1 1
D. tan9 x − tan7 x + C
9 7
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
PROPERTIES OF DEFINITE INTEGRAL
a
∫a
f(x)dx = 0
b a
∫a ∫b
f(x)dx = − f(x)dx
b b b
∫a [ ∫a ∫a
f(x) ± g(x)] dx = f(x)dx ± g(x)dx
b b
∫a ∫a
cf(x)dx = c f(x)dx =
b c b
∫a ∫a ∫c
if a < c < b, then f(x)dx = f(x)dx + f(x)dx
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#4
8 5
∫0 ∫0
It is known that f(x)dx = 10 and f(x)dx = 5. Find the value of
8
∫5
f(x)dx.
A. −5
B. 10
C. 5
D. −8
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#5
2 5 5
∫1 ∫1 ∫2
Find the value of f(x)dx if f(x)dx = − 3 and f(x)dx = 4.
A. 3
B. −7
C. −5
D. 1
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#6
1
∫0 (
Evaluate sec2 t) e tan tdt.
A. 0.147
B. 7.343
C. 4.737
D. 3.747
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#7
t
∫0
Evaluate (x − t)2 cos x dx.
A. 2t + 2 sin t
B. 2t − 2 sin t
C. 2t + sin t
D. t − 2 sin t
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#8
1
dx
∫0
Evaluate .
x
A. 1
B. 2
C. 3
D. ∞
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#9
1
dx
∫0 x 2
Evaluate .
A. 1
B. 0
C. −1
D. ∞
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#10
∞
dx
∫1 x 2
Evaluate .
A. 1
B. 0
C. −1
D. diverges
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#11
∞
dx
∫1 x
Evaluate .
A. 1
B. 0
C. −1
D. diverges
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#12
0
∫−∞
Evaluate e xdx.
A. −1
B. 0
C. 1
D. diverges
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
FUNDAMENTAL THEOREM OF CALCULUS
The fundamental theorem of calculus is a theorem
that links the concept of differentiating a function
with the concept of integrating a function. Roughly
speaking, the t wo operations can be thought of as
inverses of each other.
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
FUNDAMENTAL THEOREM OF CALCULUS
The rst fundamental theorem of calculus states
that for a continuous function f , an antiderivative
or inde nite integral F can be obtained as the
integral of f over an interval with a variable upper
bound.
x
∫a
If F(x) = f(t)dt, then F′(x) = f(x).
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
FUNDAMENTAL THEOREM OF CALCULUS
The second fundamental theorem of calculus states
that the integral of a function f over a xed
in te r val is equal to the ch ange of any
antiderivative F bet ween the ends of the interval.
b
∫a
If F′(x) = f(x), then f(x)dx = F(b) − F(a).
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#13
x
∫0
If f(x) = t 2 + 6t + 9 dt, nd f′(x)
A. x − 3
B. 2x + 6
C. x + 3
D. 2x − 6
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR



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fi
QUESTION#14
x3
∫2
If f(x) = cos t dt, nd f′(x)
A. −sin x 3
B. 3x 2 cos x 3
C. 3x 2 (cos x)3
D. cos 3x 2
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
MEAN VALUE THEOREM FOR INTEGRALS
If a function f(x) is continuous on the interval
[a, b], then there exists c , where a ≤ c ≤ b , such
that:
b
1
b − a ∫a
f(c) = f(x)dx
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#15
Find the value of f(c) guaranteed by the Mean Value
Theorem for Integration for f(x) = x 3 − 4x 2 + 3x + 4 on
the interval [1, 4].
A. 4.75
B. 1.44
C. 3.25
D. 5.60
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR

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QUESTION#16
Find the average value of the function f(x) = 8 − 2x over
the interval [0, 4].
A. 1
B. 2
C. 3
D. 4
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#17
Find c such that f(c) equals the average value of the
function f(x) = 8 − 2x over the interval [0, 4].
A. 1
B. 2
C. 3
D. 4
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
MULTIPLE INTEGRAL
In mathematics, a multiple integral is a de nite
integral of a function of several real variables, for
instance, f(x, y) or f(x, y, z).
Integrals of a function of t wo variables over a
region ℝ2 are called double integrals, and integrals
of a function of three variables over a region ℝ3 are
called triple integrals.
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#18
2 y
∫0 ∫0
Evaluate xy dx dy.
4
A.
3
B. 0.75
C. 2
D. 0.5
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#19
1 3 1−x 2
∫0 ∫0 ∫0
Evaluate dz dy dx.
4
A.
3
2
B.
3
C. 2
D. 4
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#20
3π
2π 2
∫0 ∫ π ∫0
4
Evaluate x 2 sin ϕ dx dϕ dθ.
4
4π
A.
3
8π
B.
3
16
C.
3
16π
D.
9
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
AREAS OF PLANE REGIONS
There are t wo methods for nding the area bounded
by cur ves in rectangular coordinates. These are
by using a horizontal strip of area, and
by using a vertical strip of area.
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#21
Find the area bounded by the cur ve y = 9 − x 2 in the rst
quadrant.
A. 36 sq. units
B. 18 sq. units
C. 24 sq. units
D. 9 sq. units
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#22
Find the area of the plane region bounded by the curve
x = y 2 − 2 and the line y = − x.
A. 4.5 sq. units
B. 3.4 sq. units
C. 5.6 sq. units
D. 2.875 sq. units
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#23
Determine the area bounded by the curve x 2 − 8y = 0 and its latus rectum.
16
A. sq. units
3
22
B. sq. units
3
32
C. sq. units
3
44
D. sq. units
3
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
PLANE AREAS IN POLAR COORDINATES
1 θ2 2
2 ∫θ1
A= r dθ
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#24
Find the area of one leaf of the four-leaved clover of
r = 4 sin 2θ.
A. π
B. 2π
C. 4π
D. 8π
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
LENGTH OF A CURVE
Rectangular Form:
( )
x2 2
dy
∫x
s= 1+ dx
1
dx
Parametric Form:
( dt ) ( dt )
t2 2 2
dx dy
∫t
s= + dt
1
Polar Form:
( dθ )
θ2 2
dr
∫θ
s= r2 + dθ
1
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#25
Find the length of the cur ve formed by the parabola
y = x 2 + 2x from 0 to 1.
A. 1.15 units
B. 2.45 units
C. 3.17 units
D. 4.92 units
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#26
Find the length of the cur ve given by the parametric
equations x = 3t 2 − 3t and y = 2t 2 from t = 0 to t = 1.
A. 3.433 units
B. 2.722 units
C. 1.252 units
D. 1.577 units
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
CENTROID
The center of mass or centroid of a region is the
point in which the region will be perfectly balanced
horizontally if suspended from that point.
Ax = A1x1 + A2x2 + A3x3 + . . . + Anxn
Ay = A1y1 + A2y2 + A3y3 + . . . + Anyn
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#27
A small square 5 cm by 5 cm is cut out of one corner of a
rectangular cardboard 20 cm wide and 30 cm long. How far, in
cm from the uncut longer side, is the centroid of the remaining
area?
A. 9.56 cm
B. 9.48 cm
C. 9.35 cm
D. 9.67 cm
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#28
Two blocks A and B are attached in both ends of a bar 11.3 m
long. If blocks A and B are 12 kg and 16 kg respectively, in what
distance should a fulcrum be placed from A so that the t wo
blocks will balance?
A. 3.12 m
B. 5.77 m
C. 6.46 m
D. 8.12 m
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#29
Determine the coordinates of the centroid of the area
bounded by the cur ves x 2 = − y + 9 and the coordinate
axes in the rst quadrant.
A. (1.125, 3.6)
B. (1.5, 1.8)
C. (1.125, 1.8)
D. (1.5, 3.6)
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#30
Determine the distance of the centroid from the y-axis of
the plane area bounded by the curves 9y = x 2 , the line
x = 5 and the x-axis.
A. 2.50
B. 1.33
C. 4.25
D. 3.75
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#31
Determine the distance of the centroid from the y-axis of
the plane area bounded by the curves x 2 = − y + 4 , the
line x = 1 and the coordinate axes.
A. 1.445
B. 2.396
C. 0.477
D. 0.348
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
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QUESTION#32
A cur ve has an equation of y = cos x . How far from the line
y = 2 is the centroid of the area bounded by the cur ve and the x
π 3π
-axis from x = to x = ?
2 2
A. 1.25
B. 2.39
C. 3.21
D. 2.95
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
MOMENT OF INERTIA
The second moment of area, or second area moment,
or quadratic moment of area, and also known as
the area moment of inertia, is a geometrical
property of an area which re ects how its points
are distributed with regard to an arbitrary axis.
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#33
Find the moment of inertia with respect to y -axis of the
area bounded by the parabola x 2 = 8y , the line x = 4 , and
the x-axis on the rst quadrant.
A. 18.6
B. 25.6
C. 30.5
D. 13.5
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
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QUESTION#34
Find the moment of inertia with respect to y -axis of the
area bounded by the parabola x 2 = 4y , the line y = 1 , and
the y-axis on the rst quadrant.
A. 1.333
B. 1.6
C. 0.5
D. 1.067
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
VOLUME OF SOLID REVOLUTION
The solid generated by rotating a plane area about
an axis in its plane is called a solid of revolution. The
volume of solid of revolution may be found by the
following procedures:
Circular Disk Method
Cylindrical Shell Method
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#35
Given the area bounded by x 2 = 8y, the line x = 4, and the
x-axis in the rst quadrant, what is the volume generated
by revolving this area about the y-axis?
A. 43.25
B. 25.4
C. 50.26
D. 35.12
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
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QUESTION#36
Find the volume generated when the area bounded by the
curve y 2 = x , the line x = 4 , and the x -axis in the rst
quadrant is revolved about y-axis.
A. 75.3
B. 80.4
C. 93.5
D. 45.3
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
PAPPUS THEOREM
First Proposition of Pappus (Surface Area)
A = 2πrS
Second Proposition of Pappus (Volume)
V = 2πrA
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#37
Find the surface area generated by rotating the rst
quadrant portion of the curve x 2 = 16 − 8y about the y
-axis.
A. 49.1
B. 53.5
C. 61.3
D. 71.8
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#38
Find the lateral surface area (in square units) generated by
re vol v ing the segment of the parabola
x 2 − 4x − 8y + 28 = 0 from x = 3 to x = 6 about the y-axis.
A. 103.08
B. 123.44
C. 98.53
D. 85.32
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#39
Given the equation of an ellipse 4x 2 + 16y 2 = 64 , nd the
volume generated when the ellipse is rotated about the
line x = 8.
A. 1263.31
B. 1529.33
C. 2329.65
D. 1871.67
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#40
Given the area bounded by x 2 = 8y, the line y = 2, and the
y-axis in the rst quadrant, what is the volume generated
when this area is revolved about the line y = 2?
A. 13.5
B. 20.54
C. 26.81
D. 35.75
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
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WORK
Work is a measure of energy transfer that occurs
when an object is moved over a distance by an
external force at least part of which is applied in the
direction of the displacement.
x2
∫x
Work done by a variable force: W = F(x)dx
1
∫
Work done in emptying a tank of liquid: W = γ hdV
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#41
A conical vessel of altitude 5 m and radius 2 m contains water
of unit weight 9.81 kN/m 3 to a depth of 3 m. Find the work done
in pumping the liquid to a point 1 m above the top of the vessel.
A. 166.42 kJ
B. 115.57 kJ
C. 5.71 kJ
D. 8.22 kJ
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
QUESTION#42
Calculate the work done in pumping out the water lling
a hemispherical reser voir 3 m deep.
A. 550 kJ
B. 450 kJ
C. 325 kJ
D. 624 kJ
PREPARED BY: ENGR. JOEY A. DANDAN
ECE/EE REVIEW COORDINATOR
