stpm integration solution

8/8/2019 STPM Integration Solution

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1NTECr0.Aal0N sT?/n

substif.^k.m=p)t.'

Ihelh-a6l area shoun in {|re diaqrarnsbounle{ b.{ hc currJes a=Cf,{. ;d

S- - f-l- andJ-axis. Fi^A rnis'areq.l6J

x .(r+r)

y.g> I fi,.d +he ntersec{ionpoinf .

-y,d\l -Jt

9'* ,o -v gr+l = t_t

x-L

"i=e'(- l.Ti+15??i * *= 21sx)z-2ex

Answec:

)

, f ) isoIe4 l,|'z

-3m-2=Oi ) ( rn - 7)=O

1mzZ

1 4\=7

:. <!= Laz

' r " t -(e^-t) n

td^^'*rhz)-(-

- l+t !^z+\ - - l " .2- l l

Zl"L- i n= o's3q+R

Arecr

=

Iz

( r " ) "t in

)(

J-z

- l 24=ocL

a'? o L,71s 2.74

VA\ueotx (e)- e' (e')*

Signo{$ct + 0

Sketc\of;p ./- -*

Chrrrac*erirt fhAXimqnn

=-H"-(+r]

A"=*

t-e\)c2-r (r -t ) (r+t)

a*=[(Lpl"::= f(lne)1 t" ) ' l :- ' i ' ) -

r\4)( rg\rr^ point.

K

)(

t - {nx

9e\x{".4>o: Ln' .o<)c<el]

fut lc {"r

Tha cosrAi.\a{€S ( elYraxihurv\va\,re of g

f lce\ch: \r = ln I_rv

l3l | 9=O+s3-g-+o+e=O Y=O

Acec^ f Ar I

At = (*te l , .xI

i-- r v-

ff'tn=

rynr l

-Lz

U=,tnx V=X

y*= dVrT*=

= A'*2x-l

' xi)

.t I t-t\iz--\^-. . ' --- =-- -,c2-t - : (x-r

*- .o: [ r : e '<L1

++=1 i:;=T=ry- , tnt=(

,c=)e, iic""'ar^"tis=(",)

Considtg

$=



ffi,,.fF-".,-*,1r*= tJ*^*-#n-l=

t I l,' x-') l" (x+r)\ c=;,t" f r i )*c {.

Req,.ireJo\qe

=TL+(3)'*- )

V=Vl - \

= [-f t,r"r ] [iuf*-5,1I [+o- rt(113+lgnt5-z(r lr t+i

- / I \ /-\Xt-t,;1

u

)c

'f-l

l l*=n,t +l+ -: ,H,)'1"=*Jr,r(tc"-t;'

L+"-t 2 ttL=z

,[d;,dx

+ ?l = [o(H,)-r^(H)]

uf4u;fi,a* *ulH* ?

Fi*d i*erleo\ion poi"{s.

92= l( Sqbs+\L+r @into j" =

+h(

,\+

t

=o.7+7x$

( : - r )7= ) ( -

X2-tt t (+4=)(Lt- s r+ r1= 3

Ix- t ) t r -e)=o)c=l t *"*

9=l-tt

\=x

Aceq=fr;, A9_9=-L

= _gAJ

_)Gl iy=2-X = 2-l = |)(= 4Z L^ 2-l(= 2-) =-2

-t

I

a\zlv

J- raJg=-2"

^\=l

- J (3')dyJ='>

4-?'l=+t,^t;

x t"trt"+)-3") tv+ :

x"()c"+2 ; = ttrtf

, .2(3rar-1 = lr+hV*+ ' [ttt{ '

=l

t' L

5o , =?'-+\i,::.-{J:=Pu'-'-('t-'r-?)\-t+ = 48 =(1613),

{ "t*"*rfii*t#+25'=$2jr= tlg -tL'

Y^--.-EL

u=112_ry-l

S=r-,)tnl,ilz_ _lrL

(r'-r)'

,.,4)

(+r-L)



-, -ol"[' ,{ -ojue,,"t

$=-) J=_L

=ft,;ry)'l;_:

n[ +*f:,=o[(4)3_,ry,J{i['_Ht

)

-2

&ceq =

V, : Vt: ( ' . )$

-rr I Tl:rffn vot

+23

y-6'*-sx(1, )

c\ {esq =6*-Stce6ion-boundeAy

lo[=O 1 )L- l=o)C=O t(. I

)L=Oty.Qt]-Sf =O

"=lr g = 6(t)1-S(r) :

-+( o-8): 4 -*2y= 3' ' 3 4* Filt in{s(sefionpoint s,tbst,tu}Q=?(

intoU1= "(k-'tcr)

.'

(x)'= xz (+-x.t)

Xz= ttL>-

tC*

( x=t"-(try

and 1, )

J y dx. - [*"qJrIgO 1C=O -

J'=',r-+r')d*

J*='( *-s )d,LLpO n=A

r) (

rJ'it

za

-[o

B

dK

i)(.z

J-(bt

fi

9 ,J

=GX,

D5c

)i lJrJt I

IJ

1:{3

v:ofN.=

)

X.:O

" .6I=O

r(61rJ

:

JI :O

FJ'x'-' ];=

fr.l

]i

r1t)

-n

( tc

t5-1i

,13t*sr,l;;:w?-'{Y:."

=F#'-

qi!!l-e[- [Q,','-+,1-(p)];

uni{3

Fiadal\ {hc coor/inqtes f intersectionbCtweenlhecurves5= -L2+ir and 5= Z*-*-5)c. Sketch/on he -Sa,xC

a*csr bo{{a {ll^ecurves.-ts1 Catcu(4tc {he arlq bo,^nded q+h e c{r{es S= -)c2r3x and, 1= l}-nr-Sx.tgl,(or)

e

zrl->t'- 5x r'a5z**-8rc = ui : ; * :6 I fv i=- ' } rx

x(xz-+) =o

"=Orx?=4tc=7 | ra

{=0. u:-x2+1r I l l r=o

I ri l I ' l<_ Leor<l \ rv(TL)

are(oro), lz,z1 I,-) | ^_,^\ lRe4..ice rgaAt&* B

y2gowrr s evd.n o

igLmekico\ of x-aXis

i. -2.<(Z

o- 'n.z i y, = *G:741.

, 6)"=*(+-rt) Btu=x|

(t;llY;l=2 :=r"=o1,' -2-<Y3L -X

^d (-z;to) l"-r-"- = 'i l"i 'A



the sarneoorArnl\u.a1e5,lhe,artq ,gxmi l y= ?+.3e-K.[21.Cc.\cu\rr\e(e areo.

"[ t'lîbouadtd bq {.!re- q- \

v J.. l_ l-

lE\fiîlr^[+he

coirtinataso€ i^\<sechonLetr^raen{te I1..7q= o.-+X a4A {e cu({e.{. ll-rc?.Sho.^rhat *hcof ii4texion ., {he- urrte5='t}-tt3 15a{ 1=o. t61.

l t , I f t=_6(+o)

{ ; ?-*=zt 3Cr3 =7*3d)t

€f,= 2 + Z

1=rnl IC"-3)(u+r)=o

t*i"i*-J* il ;=l t;=-',(: o t= O | ., - t.eiecf I

+3

Sketch{he areabou"ileA yt=4. cel. Carcu\r^te-.,rdco\,r-^q

"f

t*râloliA ieicratedacers rota\ca ur"ux(';ii;;

ebo,rt\c x-ax6.BT, - r | $35)

|

-&,-

4'-+t=u I t ta-Xttr_h)=o lA{Poiot{ ia{tcxioo,

j['-+I. = u0oiot o{ i

L=O tt&.tt | . ' . 6

lL = l l.=-).

' r -=! |'

r=o^l

djg*o!a ,F1py' 2-rr)t=\2

l:"?oi"tt ,.{lexioa,rtot'i=Y ,tcQ)9,12-a(")=

Yz, Fr@3"12-+[-))=loc*.rti.'"+isfO,z;, I IG,a)aÂ-2.,20)* I i

_r*,6\ttt-9=JI Ther%,ure/ ystum(,

V = Vo\urcA r l^ i .^^"$

-

vvtq ' L

"L--4t i \ yr f ,I-:f t J "

V=nl(nl tar-T{t ' _r_r=t

r :r l * . , d,.

=Tt Lt]*'+ _ finx1r't+LZ )Nq v ' tr-t

=T tq- D"l [ 1"4 r"tl

=8x is cotr^{e4omgle*c!; iti1;

Thg rcrrr*.ccl{o\.r,n, {= vJt^-rA- \q^êb

- fx.- -Z r l (=l=oJyrdr_n) g'Ar

) tao I"O

=o[_]r+r-16r+6r) lJili+-],+t'-lll

iE

-ea

'\LtEgrl 11-3J

;

i rJ1!.!9+

I:-)'

=" [**

r. - t J;;:* d. YotumeB=fft-r,)rdx ,,f-ir;6^t :O )t:O

*lr.o,=

.i"-r_*_6r)/1lc-2 ':,

,6 '

o

6z+

( *he vo\u.Y\g

\=Ag

Ralio

:Azr

AceaAr=



g"grz'-.'@ Vo\ume

$={t(u-x). . ' .

r= ^- T-

Sox=2,4. j=O

s*bs\i{,^{c,it6fr =4-oicL 2

ffiSt<et.h +h e in e q= [ aat *he cucves =8x2ondq=*.rz]:(/Co\cuta{q +he v5\ume4cnerateAwhci +he re4ion- f,i^Je,ltog he \inc anA cu(ys5 is -ro+o*eA

^boulg- a}is.uts: oosf

r)dx

)d*

=+4

+'s(+-

lf,zr

.=11

;2 0

)c=

:=Q

lr('-{ '

'r)t:

Jxt

rL

JLe (

:11

-L:+'o+t -

ctEJ

f,--o

ITt)c=o ?-U

E

J-x,--

=)

L=JlL*'

Vo\une=T\*=hqtd*,s'x,co

'=n(*- t-^,.,'., 1

,- v

-*t*irfl'

=t{-#.ftl-=' ?tr , r , i {3

15A.

dr,1"'@

- ,2q^tI

dt

drt

O= 4 -a(:)

ab-{-a=O

b-4=o

''a= - t b=

:.

1lx)2,t**=*

X7= ltx

l+tc=ox(tt-+)=oYv'-O, -=+

r=0,3Q3'l(u)'Ox'a,@ y:|(*)=1Coorrlinat S

jvot,,^.oI'7 5

_r,ly, 5''|*'i -"ll.l,l.,

=Tf- f =t-n' i ; 3 J j=o t:

5'Lr9zl t . 5=l l

=)i'+ J*i : :t 'z Y=2

=[u+#H-t+t

Y=L=rr

Jy,fr*'9=z

+Jrz

!)a, r [J'*J!*.Jl+

t

t'^l'

-r [$]v='^ [5 J5"o

[';J #no,;1

! 3 l t 'L

,f ! i ry,rJ'o ,"!io

-t-+*^l

=lj.rar-J'iars

5u\o5trtvrt(. t = +-L = + -1Linto t1""11 3 3

uottn

J

= *rc(+-*)ind coord.inatg5 { inte6

r2

)=htr- t)

& =ht-+)cz 2lL: x- 1

I = (272)a

o t,2

=T\- I i



e=IQ;--Zrt)el ttL>o

'=+6-ff ' ll = >t," Sxn"i'c L

gaJJf1\ro*3h t,-+)l ii -+=Lt)""-5(r )h+c | :- {= -4+c 11L=o l-S

Vo

Vo\u're,eL-'l=r'ly,

l, -l7?f,=L

,fi1[rt,t r ( ,=- l

'r !',6;' 153-:t

)

."X*,tuoeAm- a

t"*(t^tr)=3se +r - 3scfx i

* lA'i;=,'*. 'Xl

l"=o

=lz/,r,o;'|tLs 3 Jo

_q=, u ?.1 Ic-'au"t"1 ( 0,5) o^l t"5, )9*=^3-,sx:1

' '4x= IaS r 21=,(,r l=0

tdh-Qn\^4-$,ion is ro\c^*eA 5 cbo.*(og)

Jc'( : ',

ul -a

u.-6ur5=o I F"@)J"6-Q-(u-s)Cu-l) :o I . .=5 ,

loG- r+C)-to)= gq."'-+'dt-ft+^"Flo

' (ec+x)ai .te tSlr ;

-€c-X' 5 seL.L Hence, va,\ua

So+e: tan2lt+l=seczK)

= 3(scc2n-r)fsec'xl= 3 sec+)c-3seCx

a+ frrl= {, t[i"d f,cry. H-re,dekr^.iaq[2,((,r-)r_5 ., (1,t-5

xz .Fu=*'lv--t"-91 rs>x-1

-Ho=t*lH.=z f 'rx l=2n(n-s)r): t- /.1:ll;'""-t

f'c*t=vf 4s1-uf Ysl-rtt"9Y'l'-1--T Px-s;zt I f r , - r1 L.

fi,rd {ae vq\ue"t Sj'/iinrx s)

trtig (-t4sti++{ol

, .2

t i | [{cx1J=.cr_rr

^ n^- 1"-j) l(z ' :' ) . f t r ; f#;

_(z,t_s J 6Gf*-

: 4r2- tor - l.tt'' ^

='ti;:{; I d^=+[*i]'75;--_ I -

W=r[ r-21 r

u3

Di{r!-r4' l4)c

r .duLi^rit

-1

Fco'r@,@ u-t

,ffi'at



5anrq Sb*t ouro*o ,i..t

g!S. Jjx/r.tf r6l

ute p;r{l[*r\ichwa3 sfusfer

@t-.'-J "" ^ttr"tcia{tiub

l. .tLtrl\\t

X= 3j i \ i=,l t\3 = Lr,

-'-*!'

;{fi;t;;ffi"r

uri^J !t=

u1Dwr{ :

9wrt )L

= fzvs- 2,r3]*=zL5 5 Ju'r

r r8 loto ir

-3

I rlo,

bu

T+L-

7

,

-j e-"+c

=[r(zt1(zr3i-f+L'+lL57JL

#^*l,sl(q?s)

[!t5, -rv--rr---f/pORruU -AE \

r-,tQJfudr * e-x d,7 1.3, i lgT)

i= n a-rzli l*I

\2

2v,4. zdL-

ion eva(raa <Ji tlr-x1t

U=

&=-tJX

- du:

Fro-@ r-u€D

[?I+-"'i{ '= 2r

vtdu= '+ f) du

tC

5 t+-xt -+ cci*')

= -(h-)(z)t^*C^.

- r/4-xt+ C

dx

@

6s0. i l

l-2u

6= ,/Irt- .(zx+r) + C =,Ei(tt+r)* 6

6-___7- '+(/rJi^3+hCaggroriateeuros

J HAx.rsl(or)

=\6",-z) "=fa:d-xu]+

= 2u?-8u + C

2(,1ffi 3-SJrrti+ CUrig he s*bstifulio,ru=3+2siq0 , euq1.$ot( t/ e cosO

-o gjt)s;€J,

dos . dOcs: 03) c."e4recae+tctA4t

"t xz

3+2rl,re)2 (.o*f,.R cpc{{\(iaatof [ ,

-24+B=O->(t1+B 2 O

-'.B= 2.'. )Lz_ _L* z * I

e-,)j ' x-r'[r1r (rt-r1e

961 ors)

=$+ X,,La6

ffi!f'= ox----Jff iJf 'fn=

{n*'

,u ' 3+15' .n{-4

Jt{ ,

u:3+)r i^o:3 =iL-i-e!t j=



Qt"+t)(1-x)

(xz+)= )r +

= i^fu@t^

-,l=

dxhl @Jahr-v1o+)

}(

r+e=\*#"l=[ ta*+] f Ja

=Jr )tE.),1*

= -)rd'{rt')+ [e:(**'),1,.= -lg-(xtt; _".-Lx<t) 4 a

&r;.tt''rc,$Jftfr^*Fa*) ,nnrw(r:

ffif15l llcnler

dtfe (^'t)zlx= J,"(nrftx)1

ft^l Jf (rx+r) a)tdx. Give 'he rsr,,cr 4c

nz ( oL , t r"* ] ,

7 - ti- tJ=\,e+ (3a.1.1

4,1d*

v,JtJrl

!f., IV"*2'(e,tgr+l)+9*]

dvdL

{:

: ql(r+t)

Id('*']*

ryD- l

stpm integration solution

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