CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 2 Examinations 2014/15
Module Title: Mathematics 4 (NMCI)
Module Code:
MATH 6036
School:
NMCI
Programme Title:
Bachelor of Engineering in Marine Engineering
Bachelor of Engineering in Marine Electrotechnology
Programme Code:
EMARE_7_Y2
EMAEL_7_Y2
External Examiner(s):
Dr. Jeremiah Murphy
Internal Examiner(s):
Dr. Tadhg Creedon
Instructions:
Answer all FOUR questions. All answers should be displayed
correct to five decimal places.
Duration:
2 HOURS
Sitting:
Summer 2015
Requirements for this examination: Mathematical Tables
Note to Candidates: Please check the Programme Title and the Module Title to ensure that you have received
the correct examination paper.
If in doubt please contact an Invigilator.
Q1. Evaluate the following integrals:
x3
3
 ( x  1)( x  2) dx
(i)
(8 marks)
1

x
(ii)
3
sin(2 x 4 )dx
(8 marks)
3
ln xdx
(9 marks)
0
3
x
(iii)
2
Q2. (a) Use the tables as appropriate to evaluate the following integrals:
2
(i)
10
1 9 x 2  25 dx
(8 marks)

(ii)
 sin 3x cos 2 xdx
(8 marks)
0
(b) Calculate the mean value of the function
y  x cos x
between x  1 to x  2 .
(9 marks)
Q3. (a) Find the root mean square value (rms) of
y  1 cos x
between x  0 and x  2 .
(8 marks)
(b) Consider the region bounded by y  3x 2  2 , the x-axis, x  2 and x  3 .
Calculate the volume of the solid of revolution generated when this region is rotated
about the x-axis.
(8 marks)
(c) Consider the region bounded by y  2 x 2  1 , the y-axis, y  1 and y  3 .
Calculate the volume of the solid of revolution generated when this region is rotated
about the y-axis.
(9 marks)
Q4. (a) Suppose y  f (x) and that the following values for x and f (x) are recorded:
x (horizontal)
f (x) (vertical)
0
12.5
2
15.8
4
19.7
6
25.9
8
32.1
10
46.8
12
32.2
14
21.6
Use the Trapezoidal Rule to estimate the area of the region bounded by y  f (x) ,
the x-axis, x  0 and x  14 .
(10 marks)
(b) Two cars A and B are being driven side by side at the start of a race. The table below
shows the velocities (in miles per hour) of both cars during the first 10 seconds of the
race. Use Simpson’s Rule to determine how much farther Car B travels than Car A
after 10 seconds of the race.
Time t (in secs)
0
1
2
3
4
5
6
7
8
9
10
Velocity, Car A (in mph)
0
18
30
44
52
60
69
73
79
85
89
Velocity, Car B (in mph)
0
20
35
50
59
70
78
84
92
97
101
(15 marks)
Formulae
b
Mean value =
1
f ( x)dx
b  a a
b
r.m.s. =
1
2
 f ( x)  dx

ba a
Volume of solid of revolution about the x-axis:
x b
V
  y dx
2
xa
Volume of solid of revolution about the y-axis:
y b
V
  x dy
2
y a
Trapezoidal Rule:
If y  f (x) and we have equally-spaced x-values: x1 , x2 ,, xn and corresponding y-values:
y1 , y 2 ,, y n , then
xn

x1
h
f ( x)dx  { y1 2 y2    2 yn1  yn }
2
(where h is the distance between adjacent x-values).
Simpson’s Rule:
If y  f (x) and we have an odd number of equally-spaced x-values: x1 , x2 ,, x2 n1 and
corresponding y-values: y1 , y 2 ,, y 2n1 , then
x2 n 1

x1
h
f ( x)dx  { y 1 2( y3  y5    y 2 n1 )  4( y 2  y 4    y 2 n )  y 2 n 1}
3
(where h is the distance between adjacent x-values).