Test 4A Solns
5
Math 80 Test#4A Ghapter 6 Work arefully. o u must show all work n a clearly rganized manner n the space provided n order o receive futl credit. 'gXnCf ANSWERS NLY, nless therwise pecified. a ch roblem s worth 10 points. 1 . Setupandsimplifytheintegrals ecessarytofindthetotalworkdone y F= (2x+z,3xy+z,x+f) naparticle thatmovesfirstalongthelinesegment 1 rom 0,1,2)to(2,1,3) henalongthecurve zift)=(t*t, t,t2+Z)from (2,1,3) o (4,3,11). 0 ' \ ' zC cJx= d t , v f 'y=l drl=O tl)-rl L=Ztt di=dt $ rr*tz)dv (sr,6fz)dl ( v y)dz ('t4t+zrd7dt)t l Qttr)(dt) J o ' : \ ' (torr't z*Det = t J f t z t t5)dt ) o C z : F= ttt, , *+z\ [l;lli] J = ' 5 , 1 ztrZt rz)dt + . 1 ' 3 9 z)ot t(L;i'i v\tldc) - 2 l ' f : (tr1i-to +tru+r'r!! l?4e - \ tt')dt TorA t,voLv-- 5 : (att5)dt l i (qt't-ltlb) d k 2. Evatuate he ine ntegral I x]cos{z- x)ds where C is he curve (f ) = ( 2, n(cosf , *r)jo., .! : 0 -+ 1 7 t e c r'=(o,#, r> I . ' r{c+z5c$\e) a + , ^=<o, 6a6f , r ) lt {- ll1=.J]offi=s€.t = ? \ ( e z)dt ) c = lu.df I , = ?(Lt-e 3or) 4>
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3. Consider thevector f ie ldlrJr*a *2xev,xJz+4+x2zew,xy'
*x2yev' \ '\ ' 2Jz+4 |
; lv
r.{U
*Z-r
*ar"r^"rl^rd$ry.ar; (-, -4) L wt"4)vtt pV cla,/\prj<-\ oa\^d\tt' Nt't\t a,!z
a/-eM oA\A dS"Ah//\ (L,,u)=6, F r, c,b4€,qvoj-r).v-
= (uro,.:)
b. Find hepotentialunction.^
4,"$z r.=S{z14t
u
n - xt tf -tb
Zl -Zfq
J,f.Y#6t
I f1e1z
r fd*Jh(u=xg.lfi tr-.$'+$gO f=Uffi +f#+h(Ye)
' z ' Y zvqvt+ t \c
c. EvatuateI[tr:,2'r|,t,.+zxe.)ax(rJ, *a +x2zew)ar.h* r'v"wl,
14,1,5)€(3,z,o)
f=
' :3 t c 5 - 1 2 - o l
#.jn ." r(u,v) /2r"o.n,r, nv, 2)indhe quationf he angentlay! the urfacet(A"'8J,4\-,0
G= (Zc*r, srnv, ZtD
Fu= -2uanv,L^.6v,O)
vr--Z ')ctnv= 2G srny
C6vc{g V=
G, ,<)o)
n =(- 7-(3,-t,4) r U,tot - / - ; \
G(x-zfa),\-r) -r,- 1):- z t t - - o
rftfr ={; i;- t=l . ; G " l s,141 (",n, 2.,
lJ-gXZ
- i : +- - l
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S(&+2.\x{co'fu)d/H
IS3dA3G"*^%*;/")
6. Find hesurface reaof hepartof heparaboloid= x2+y2 with 2 < z <6.- =+- *
:
, . a t
d5='l(z*\'r(zrTI dA=rl4f t4tit r d,A
(+r+,)u'
S. UseGreen'sheoremoevaluateJf
.Arwhere =("t sin2y3x+2excrrs2y)f C s hepositivelyriented
c
boundaryf x2+y2=25.
-. [4u'11dA
6r a,lb
\\'[+r+l v^ ,'dAc
*''rft-
^{+.t r:lz al-z 54'-? t,1-C_1A-
J : *[,zs l=fl{rd-
3(r)(5Y
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7.
I : (,6rW)
ffi)&rd{I (+*
t | %dx=\:@q nr.r[o)r*Ift"u rr-Grldx
=( (u,fzuzJdx= Gt',r-trax[]
Findhe ux rFthroushrF= 0,"' ,0) ano,.,ili1',lriilJ"CXt Zwith1<y<band riented
towardshe ositive-axis.15-r
.; cls 6 = ( -Zy , t :ZzJ= ( - Z y , t ; 2 2 ) l
II rdAb
7'l{s^zrlll+[]\ :
\ \<",f ,O.?*,tF=) dAb
) r l+x'+--b
x'taz =57
f..3saott.*h ae
2
f zTf Gr
\. )ott"3e ' r d-,^dt
(* r d,'r
) D
d,fr @
1(-6{z rra)
1q-zlQ
- (rr.X?)
fd" c
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9. UseStoke's heoremoevaluatell"qf_yAlf .=!u,?!, x +z) whereS s hehemisphere2+y2+ 22='l ,
*,,nffintedupwardr*, \F.d; l.| c iv=.f'= l,z=o -t1-- cost,Jnt,o>+ 4)F'--(-srnt,-** ' , i l . t ' t )Azf Y
-ea\\t")= 3T
'10. EvaluatelJr.nos
when =(xy2,yz2,x2z)noS sthesurface f hesolid nclosedy hehemisphere
Pade
=JTL-cost,lJf(J]5:
zi f t ,1 [ '+ ]
lslpT5
