efa1103 test2 - solutions
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UNIVERSITI PERTAHANAN NASIONAL MALAYSIAEFA 1103 ENGINEERING MATHEMATICS
TEST 2 SEMESTER I 2013/2014
DATE : 25/11/13
DURATION : 2 Hours
LECTURER : Noor Hafizah bt Amer (1JKM1)
Syafawati Hasbi (1JKM2)
INSTRUCTIONS:ANSWER ALL QUESTIONS
NameMatrix No.
Course Outcome Questions Marks
CO3
Apply appropriate procedures to determine integral in Cartesian, polar, cylindrical and spherical coordinates by using multiple integrals.(C3)
Q1 /10
CO4
Solve various problems in scalar and vector fields by using appropriate theorems.(C3)
Q2 /10
CO4
Solve various problems in scalar and vector fields by using appropriate theorems.(C3)
Q3 /10
TOTAL (15%) / 30
TEST 1 EFA 1103
QUESTION 1 – Triple Integral- Cylindrical coordinates
Using cylindrical coordinates, evaluate the integral
(10 marks)[CO3-PO2]
QUESTION 2 – Vector Functions
A spaceship is traveling with acceleration
At t=0, the spaceship was at the origin , and has initial velocity . Find the
position of the spaceship, at .
(10 marks) [CO4-PO2]
QUESTION 3 – Scalar and Vector Fields
a) The temperature of a point within a region is given by the scalar function
. Find:
i. The rate of change of T at a point P(1,1,1)
ii. The rate of change of T in the direction of the vector
iii. The magnitude of the maximum rate of change of T from point P.
(8 marks) [CO4-PO2]
b) Magnetic flux density of a point in a magnetic field is given by the vector function
. Determine whether the function
is solenoid or not.
(2 marks) [CO4-PO2]
P a g e | 2 Semester I Session 2013/2014
TEST 1 EFA 1103
LIST OF FORMULA
Identities of Trigonometric Functions
1. sin2 x+¿cos2 x=1¿
2. sin 2 x=2 sin x cos x
3. cos2 x=cos2 x−sin2 x
4. sin2 x=12
(1−cos 2 x )
5. cos2 x=12
(1+cos 2 x )
Double Integrals
∬R
❑
f ( x , y )dA=∬R
❑
f (r cosθ , r sin θ ) r drdθ
Triple Integrals
∭G
❑
f (x , y , z ) dV=∭G
❑
f (r cosθ , r sinθ , z ) r dzdrdθ
∭G
❑
f (x , y , z ) dV=∭G
❑
f ( ρ sin ϕcos θ , ρsin ϕsin θ , ρ cosϕ ) ρ2sin ϕdρdϕdθ
Scalar and Vector Fields
P a g e | 3 Semester I Session 2013/2014
TEST 1 EFA 1103
P a g e | 4 Semester I Session 2013/2014
TEST 1 EFA 1103
P a g e | 5 Semester I Session 2013/2014
TEST 1 EFA 1103
P a g e | 6 Semester I Session 2013/2014
TEST 1 EFA 1103
b)
P a g e | 7 Semester I Session 2013/2014