Universiti Teknologi MalaysiaDepartment of Mathematical Sciences

Semester L 2OL7l2OL8

SSCE1993 Engineering Mathematics 2Individual Assignment

Topic covered:

Chapters 3 to 5 (refer to Course Outline)

Instruction:

1. Answer all questions (1-4).2. write the detail of solutions using 44 paper. cover (front page) of your

individual assignment must contains UTM logo, your name with matrix no. andlecturer's name. Compile all papers together and stapled (not need to bind).

3. Submit your assignment

QUESTTON (1)

(a) Find the mass of the plane region bounded by the graphs x : y2, y : X *3, y :_3 andy:2 if the density is constant,

(6 marks)(b) Find the volume of the solid cut out of the sphere *' + !' I z2 :4 by the cylinder

x2 + yz :2y.

(7 marks)(c) Evaluate

I$a7;r*where 6is the solid bounded above the cone z =x' + y' I z2 :4 and below by the plane z:0.

, the sides by the sphere

(7 marls)

x'+y'

\

QUESTT0N (2)

(a) Given the vector field | : ylttx i*3zstny j * e-,, k , find.

D div F,ii) curl f ,

iii) grad (div f )[6 marks]

(b) Given r.$):(3sin}t+st)t*(3cos2r #)t.Findvelocity, u(tlnaacceleration,q(t)atthe maximumy-values of the curve.

(c) Given the scalar field[7 marks]

Find the directional derivative of S(x,y,z)at the point P(3,0,4) in the direction ofg: L+ l_*!. Hence, obtain the opposite direction and the minimum change of

0(*,y,4at the point P

[Tmarls]

QUESTToN (3)

(a) Evaluate

Irt, + z2) dx + ry' dy + xz dz,

where C is the line segment from A (I,O,Z) to B (4,I,3)

(b) Show tfr"t ..[ f .d7 is independent of its path, given that[6 marks]

F x,y,z : (6xy' +222)L+9x2y2 j +(4xz +1)lg.

Hence, find the potential function of Fand evaluate its work done to move an objectalong the line segments from Z(0,0 ,Z) to p(2,0,I) to W(_I,2,-3).

[7 marks](c) Use Green's theorem to evaluate

('!.Qt -t2xy *tnx)dx*(3x2 *3*y, +sir/r.y)dy

where c is the boundary of a region bounded by the straight line y :21x , the

semicircle ": {1- (y -1)', and the x-axis in anticlockwise manner.

[7 marks]

x, +y, +2,

QUEsrroN (4)

(a) Evaluate

f,F.*by using the Stoke,s theorem given that the vector field

F x,y,z :(*, +y, +y)i+2zj+x2k,and the closed curve cis the intersection curve between the pranez:2 with thecone z: transverse countercrockwise as viewed from above,

[Zmarks](b) Use Gauss'theorem to evaluate r t,

J J.E'ryaswhere E x,y,z :(r+y')i+(y+zzh+p+x2)L,

oisthe closed surface

consisting of ellipsoid 5 *5 * *t: 1 and n an outward unit normal of n.

(c) use stoke,theorem ,o "r",u"I" . _

[7 marlcs]

J J vxf .uds

where E(*,y,): xti+ x+e"' i*ru.k and cis the intersection curye betweenx' +9y' :9 and z: x, transverse counterclockrrise as view from above.

[Tmarks]

i1

quEsttoN I Czo rrrqrzr,s)

q) lprnd Jhe ro qss of .{be

rt

plon0 ce

9=) ff

bounded

{hg densi

th4 nqph S

ti cs n.ff qnt . f 6 mq'rs l2=Ut, V=I+3 = -3 ood

fhqSS o{ plor,O rOgion >[:J'

Y= )t+ 3

_) | drdg

rxl:-. d.y

( s'- v+l ) dY

lo7, v7, + 3y J-,G'/l + r> V,

r+s/5

mqss = f ' A tl's /6 p

Y= -3

V=).

rrqls o{ elqle rggioo -- p C deo$te) x areq o+ re gion (A)

x+3

x=V t

A: ff JI I drdy

J:v

Rab.[o*

QUESllbh,l 1 - contrsee

(b) Flnd +b € volvm4, of lhe so lrd cqt qut *ho he,rg xt+ 9t+?'=4*u,g Irie. xt+ t=29

N5\5w9c :

t :..:- +-)t'-g'

sAx-+ u

a=(cos9 , 9'- c SroO

rt2 + (,y-t)a=t

rtcos'g a(rsrng-t)t =1

F: I Sri9

g ) 5rn9 .lliT'I . r dzdrol9

4- ra

rLzl drdO

f? tno.rs ]

+ :Lt+yt-2-9 =or("+ (u-l)a= \

cenkg (0.t) ,F=l

rr J 4-r' dr d9 - \e*u=rr 9&{=lr'dcJ"n

(4-u)''' dq dB

]"

JJI

d9

-V, (4-r')

\.tr.86qt L0l;

. 1" dg

Lrf

J

,/t

)I

\6

4 r)3',>

dg'/ t c+)

4 -r'

Rbtro*

1.86q) F x

+ ')de

QuESltoN t - con*inqe

EVq\uqk, &V

i-whgre q is {\e sot,d bounded qbove lr\o cone z = J xz+ V} , *he ideS

by {he sghece x>+ yt* L2 = 4 qnd beltw {he plone ?=o' [. 1 ..o.t s ]

a'l '+z'= *./= z- SC:rJ,z) + I Cf ,O, O)

1: l^ coSO

g a p srr,e

z=Pco9Q| ,p sr; 0

*=PSrnpcoSg

xt+ t+ ?"= ri=p.,tiPsrrigdv . g'.,i + dp d0 dg

l$z 4-x z dzd dr*t*y\ z"

cos Cl cos O

x'+ y" + ?'

z dv= coS S ) str. 0 dpd0de

f'

x1y'+?'

=J L i ].,' cos 0 sr'i q dQ de

+ cos 0srnAd0dg

Rob.[o*

O by s\,rbshh*,bo ,

te* U e Srnd du=qsfdcldtt

= cos O IQ= du

cos p

du = cosV d0

- r5; q,kA cnsPdsr/4 q, dq &o

I u7,: J:Ttrt+ dg

srn t

:" '4 es

e/\

d0

r 2If> l,r

=lT

or o:Srri rS dO dg

- cDs 26

* srri r0 . :fro 0 cos /:nt

=f ( - cos

sg2

-_ fI *

W-^l'ou,

o -tn,n

-ni= -3Sm *a

-na-\e-

t

'il dtvT : v-,F = 1-> Ch )z

i

rf

',4LHtffiti-

\

(b) r(+) - (gsrnau +gr\ r + (gcosrL{r\'r \v lt\/ I ,,

\v t+) = r

tt{) - ( r. crrs It { g \ r - G srn(rt)'r -ffi\6ftv.rv-\ ln L 1-z \/

q (+) = r" t+) = -lr stn rL le cos rL 'r -GrG)rw d/ JA

tIlnx rl - Val\lLJ L, (ft

V(oJ : qi r\.|k

5\rG'rg(o) = -l)'1,** (r rrxrrs )

C \ -11n, a+aA\rnf, rJ d' 'ts V A$ I t

= /- nt rt -+ \ -6i\\ (r" tti'1g'fl' (t'tn?+t')'ft (rr'-{u'{ffl'

5 -(-J+(N -lca +{- dnrceAon- cl ( r . r, t

\s d -

--+9

-v'r0N)

1- m\n vart<- 4 rr,,,arar- -NOll = - lgp)t 1#/") = - l -(o

E.vql\4qk Cy+z') dr * )Yt dy * "* o*

wher.g C e hne sggn€o* frurn ACl, o, z) +o B I r+, l, ]).

L+hri | 6 navksJ

oA.+Ib

-)

->

k).

+

*

L

i)

6 q t, o, z) t=ot=lJ.+!

+ nu (t)

b (4,t, l)

r C*) = oEt

= (t +3t)i + tj r C>+ t) L

),< t+ 3t dx= )dt!: t dy =dtV= r+t dz=dt

C y+2") d x xytdy + xz dzt

fT^

ct+ ce+t)');dt + Ct+3t)t'lt + ( r+ :t ) Cr11; 6g(t + 4 -t t'-r +t): + tt+3t3* 2 r-t+6t+)t1 dt)t3 + ?t'+ l6t + t+ dt

t-]t* + +tr + t6t'+ r+t l'+3)-

3l +7/l+g +r+)37 or 18 or^ )8.0 &

t2

Bzb,lo+t

b)

qUESlloN 3 - con*i".q€

Show +nol dc lL rndEerdent of i+s pq+h , gieo thqt

F[x,y,z)= ( gxy]+ tz')! + 9t'y't * @xz* r) E

llQncc, find -lr,e pufenrrr'at fooc+,o,t ef E scd evalqqte r*s work done fo

move on 0 \ec+ olunq *\g lrne vb€o{g fn$ V(0,0,:) *9 P(>,ort)

to w C-1, ), -)).

$ <4r,: + r) - *= 9 r'y') I (fr r+rz * ')- +(6ry)*>1';

+ (* 9 xty' ady

( 6ryr* )z')) !

= (, o-o); - ( 4z- +z)L t c tg xy'- tSry')!

oL- oj +o! l< ini e,4ed en* o( ? a+h

Y0 =FO* = Gxg)+ >z'=) Q . 2 + l-X,z'+ cr

[1 trort s ]

e f. r.or

Q,l = grtyt =+ A:l, = 4tz +i + 0 =

a)4x+ +z t Cr

{ rt' + c-..

d = )xtyt + )xa'+ z -F c

worfcdooq': f. E. dL = dw 0v

= Q(-t,2,-3) QCotor2)

) (-r)'(.r)r * :[-r)(-l)'-; + c- (z+c)

>+ - 18- I - z

6lyr+ rrz 9r'y' 4)tz+l

Ra*.[o*

QLlEsnoN 3 - coh{iru€

Vse kreerl'S *beorern *o eVoluofg

C qt + 2xy + ln x) dt + Cax'+ 3r'yr+ sinh y ) ty

where C r's 'th e boqnd o{ o rs gion boqnded 1t^'e s*rori hi hri e

= )* X +he senicrrclr )t = J {)1 , qcta {he >(- q?cis tril-tLr marKSlorn{icloctcwUe h^oobgc.

9" (y'* >xy+ lr'x)dr+ C)x'+)xy'* srnhy)dy

( ex + 3y') - (lyt+ tr ) ds

{iEF4>L dxdY

-f-)

-2

r [x'] dg

l- (y-t)z- (9-z). ] dy

(-.y'*6y -$) dY

- 2.Y3 -r 6yt - +y

Q:t=61 +3ytPy = 3y" +2>C

Rz.h{ro'u,

: -$

QUESTIoN 4 Czl norks)

EVqluqte

by uSr,i 9 S*ohe 'S *\ eo rem grveo {\ q* {\e vec*or fi e ld,

= [ x'+ gt+ Y)i + >ui + r'E-

f, F- u;

F t>t, 9, z)

qnd +\e closed cqrve C tS +he rnt ecs ec*ioq c\4rv e be-lween {t"'e plqnq

?=2 witt' the ConQ t = J-xnqt ir oosverEe corrn*erclocltwrfg qs

I n norks ]viewed {om 0bDVe.

LLPnnlwsf :

s \abst'"f,rie VXF . A =1-2,-2\,-2Y-l>'<O,o,l)

= -2y- i

-29- I ds

(-:tFstnO)-l)c dndg

[-trrsrri g- .l dndg

p3 srn g

I.t=k= \

. = J xt+g'

* xt+g'= 4

o<rsz o<g<2Ir

. Jo* Jo'rt lz2

d9

StnO - )' d9

-22 cos o a0 l"_ r/, ( cos urr- cos o) - r Czn) I-4t x

, e"o.=JJ vxF.n ds

= 5f vxF.N dA

'/r* v>v

ft+y'+y )z 1r'

*ri- (zx+o)i -c:Y+t)E-zi - :xj - (zy+t) E

Ro*^.[o*

!'L

=ZQ

l-do

ia1= -q du do

ort { il-{ +

&&Hass,Al

-l du =r\sfr1r

dr - dni

'\ (-rl cos3t

etd lJ+

- Bl ctse t srn { s<,os

8l csse t srn L

3t s

u

{lco-:a ! -< all

rrt

0

aT l8l\

du

k+

;5'

er

b

9=

a9>

8l ctrst t 3 sm(at) -t sm +a +

4l-HAseti