8/10/2019 MAT355 OCT 2010

1/4

CONFIDENTIAL

CS/OCT 2010/MAT355/431/455

UNIVERSITI TEKNOLOGI MARA

FINAL EXAMINATION

COURSE

COURSE CODE

EXAMINATION

TIME

CALCULUS

III FOR

ENGINEERS

SERIES

MATRICES AND VECTOR

FURTHER CALCULUS

FOR ENGINEERS

MAT355/431/455

OCTOBER

2010

3 HOURS

INSTRUCTIONS TOC ANDIDATES

1.

This question paper consists

of

five

(5)

questions.

Answer ALL questions

in

the Answer Booklet. Start each answer on

a new

page.

.

3.

Do not bring any material into the examination room unless permission is given by the

invigilator.

Please check

to

make sure that this examination pack consists

of:

i)

the

Que stion Paper

ii)

an

Answer Booklet

-

provided

by

the Faculty

DO NOT TURNT ISP GEUNTILYOU RE TOLD TO O SO

This examination paper consistso printed pages

Hak Cipta Universiti Teknologi MARA C O N FI D E N T IA L

8/10/2019 MAT355 OCT 2010

2/4

CONFIDENTIAL

2

CS/OCT 2010/MAT355/431/455

QUESTION 1

a) Use any relevant test to determine whether the following series converg es or diverges.

^ 5

k

+ 2 k

? 3

k

+ 4 k

H)

2 r fT7

i ) S

fci 5

2 k

(3k)

2k

w ln(k + 3)

(14 marks)

00

(1)

k

k

b) Determine wh ether the series V -^ - converges absolutely, converges

U k

2

+4

conditionally or diverges.

(6 marks)

QUESTION 2

( - 2 )

k + 1

( x - 1 )

k

a) Determine the interval of convergence for the series ^

U 4

K

(k + 1)

(13 marks)

b) Without evalua ting the integrals determine if the following statem ent is true or false.

Show some w ork to support your answer.

i) | x

2

dx + ycos(2x)dy = Jf 2ysin(2x)dA

wh ere C is the boundary of the region D with a positive orientation.

ii) [ xy

2

dx + (x

2

y + e

y

)dy = f xy

2

dx + (x

2 y

+ e

y

) d y

C| J C2

where C is the line segment from (1,1 ) to (4,1 6) and C

2

is the curve

y = x

2

+ 1 from (1,1) to (4,16).

(7 marks)

Hak Cipta Universiti Teknologi MARA CONFIDENTIAL

8/10/2019 MAT355 OCT 2010

3/4

CONFIDENTIAL

3

CS/OC T 2010/MAT355/431/455

QUESTION 3

a) Con vert J [

x

f

x y

(x

2

+ y

2

+ z

2

) d z d y d x to an equivalent integral

in spherical coo rdinates and evaluate the integral.

(10 marks)

b) i)

>i)

Show that the Jacobian transformation J of u = , v = xy , is given by

x

vU Vy 2u

Use the transformation in part i) to evaluate

f f

R

x y dA

where R is the rectangular region in the first quadrant enclosed by the lines

y = x , y = 3x , xy =1 and xy = 4 .

(10 m arks)

QUESTION 4

a)

b)

c)

Evaluate f f 2 e

y 2

dydx by first changing the order of integration .

JO

J x

(6 marks)

Use an appropriate coordinate system to evaluate f ( x

2

+ y

2

) d A where R is the

R

region bounded by x

2

+ y

2

= 9 .

(5 marks)

Use the d ivergence theorem to calculate the flux of F across S where

F(x,y,z) = x

2

+3 x ] + ( 2 z - y

2

) k and S is the surface of the tetrahedron bounded by

the plane x + 2y + z = 4 and the coordinate planes.

(9 marks)

Hak Cipta Universiti Teknologi MARA

CONFIDENTIAL

8/10/2019 MAT355 OCT 2010

4/4

CONFIDENTIAL 4 CS/OCT 2010/MAT355/431/455

QUESTION 5

a) Given F(x,y,z) = e

y

i + ( x e

y

+ z

2

) j + ( 2 y z - l ) k

i) Show that F is a conserva tive vector

f ield.

i i) Find a potential function f(x,y ,z) such that F = Vf.

iii) Hence, evalua te f

F

d f where C consists of the line segm ent from (0,0,0) to

c

(1,1,4),

followed by the parabola y = x

2

from (1,1,4) to (2,4,4).

(12 marks)

b) Use the Stokes' Theorem to evaluate j V x F n d S for

F(x,y,

z

=

(e

z

) l + (4z - y) ]

+

(8x

sin

y) k

where S is the part of the paraboloid z = 4 - x

2

- y

2

that lies above the xy-plane.

(8 marks)

END OF QUESTION PAPER

Hak Cipta Universiti Teknologi MARA

CONFIDENTIAL