Universiti Teknologi Malaysia Department of Mathematical Sciences Semester L 2OL7l2OL8 SSCE1993 Engineering Mathematics 2 Individual 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. and lecturer'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 and y: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$a7;r* where 6is the solid bounded above the cone z = x' + y' I z2 :4 and below by the plane z:0. , the sides by the sphere (7 marls) x'+y'
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Universiti Teknologi MalaysiaDepartment of Mathematical Sciences
1. Answer all questions (1-4).2. write the detail of solutions using 44 paper. cover (front page) of your
individual assignment must contains UTM logo, your name with matrix no. andlecturer's name. Compile all papers together and stapled (not need to bind).
3. Submit your assignment
QUESTTON (1)
(a) Find the mass of the plane region bounded by the graphs x : y2, y : X *3, y :_3 andy:2 if the density is constant,
(6 marks)(b) Find the volume of the solid cut out of the sphere *' + !' I z2 :4 by the cylinder
x2 + yz :2y.
(7 marks)(c) Evaluate
I$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