Dept. Dr. 2*class

Al-Mustansiriyah University/ Faculty of EngineeringMechanical Engineering Dept. Dr. Hayder Mohammad

Mathematics Lecture

2*class

D oqb /e ru/s ./o,- Co-o-vJi-w

the ovt

I

Ioa

A fo/*r coorf i.",rri-+ o ca4eJ.[1"9 Seg m€'tt Sefrt^L S.',,^ th ec44J (L,e [>o(a( d.x;I \

Ro Jio( axis

,PCrte)

t go0or c)xt'S

f - fa.olt'o.( coer",ti^ #a*- {>a /"r o*g^.(<

po(or @oy"oh'vate( ur|. a. foi*t f r5 wvl#fs..ag Che) where r ii\ t/,e o.lrl*"^ce .{ Pf*o*tLre po(e a^d g ,"S €he an.I/en%sur.i f*^ tLs [>ohn ax ir"a tL t-2,'n, oP Cno.Jr,ry' a,x,'S)

(


Mathematics Lecture

2*class

1

or

x

/= Y-CoS, *

8 = \|: Si'^&t{x t: \r := io.'e*)

xI

Let R be a. si*P/c y'gtl\o^Who gC b 0.,,..^ ola-y-ic S afe €h e fa"l S g=4-or^d g= B q." r( Curv6 y - giq ana{,r = ?"Ce) r f f f Ch?) ri covrfl\utotS o\4R, 2 t|e,^\ r= Z.Ce)

{ox\)dft - f cnr&) 'rc/ Yol&

fu=\ f=gt@.)

5P= et

9z-(e)

= %\ftT>X

f

L

r

PCrt*)

e=g

e--F


Mathematics Lecture

2*class

ftaudar es( (r 17 rll

T- For,^n-(r'es "+ cfrq/s1

@f=c(Sl*,^&

: A,CoJ d_

@ I^5- qSi,"<9

@ Y-= - aC"5*

t1

q

X

X

u)

v

xq

x

q

9

3

O y=^ q

@

qA

Al-Mustansiriyah University/ Faculty of Engineering

Mechanical Engineering Dept. Dr. Hayder Mohammad

Mathematics Lecture

2*class

)

@ V^^ q. crs n& g

I

z /=_p <-/ -r,, ) rt n L) ,( t 5l

vF - ft foc(>,O- ) )I _ yr

-J)\- "/

E- ft rcre Curve

O y^= a 6lr^C n 9)

x

^

n'-l

A

4

Al-Mustansiriyah University' College of EngineeringMechanical Engineering Dept. Dr. Hayder Mohammad

Mathematics Lecture

2*class

X

fr,r^ oS),^(3c-) 1

[= a,CeSCSe; )

\

Y= aSi\^ zcY

/

)

x

\

x

5

Al-Mustansiriyah University/ College of EngineeringMechanical Engineering Dept. Dr. Hayder Mohammad

Mathematics Lecture

2*class

Tff - F.,t\^,\i .ltp N Lery\!^5ccfeo

e (': orzSr'.a Czg)

?.JL\€ r.,t c, = b erca ile-J Carolt'oic,l beCatesC a

Y

x

x

thc< CarS tS{ r'€s tlrle

)

f 1|- Fo vî (ies "N Cqv-dt'oi ol q6ol /.,'.o cotrs

n\q+bcos&y^= a, + lD Sl\,\&

Po 1ar Cu t-V € t' La t.;t^.t'c L^ ct) o u,ôl b ).

qb

\ L

Vurt SLre pn"l appe*rq,^ce s o :

t

C r|2= q?c.SCzc-)


Mathematics Lecture

2*class

)

Y

x

f=a.Cl -Si*.r*) v

+

( = a, Cl+ C, s & )

f ^ aO-CoJ@-)

y'= o Ctr Si\^ cY)

/\

x

v

A


Mathematics Lecture

2* class

SXty F*c/ t1.e erea,b g fike {n,o crrcqbr

S o! urtr'.s.*-o-

" { (ke r.?'o t-\ boutnde/

y^ s C-e3Cq o;r{ V^=Si^& \

v\r

e+ r S

V?CO9 &/

e.E ti c.s*

n r olv^

O=a 1=O Q=pc{

Y-= o

Y,/,, S*P.l )q r g-t" |i\;L Lt' Ct a..r t Y-l1AS\ ia\Jt ;w.Y 2 i

3 D,ei

l

rolro@ +

Ec/ \ &

fr L rd(d&= 2Ir+.ra.<)

o

J&

5ir^<& r)*o

!z

I2

b

o

Ifr

0D

A1-Mustansiriyah University/ College of EngineeringMechanical Engineering Dept. Dr. Hayder Mohammad

Mathematics Lecture

2*class

D

"al- + (e-ISi ta

L="f

-T,s=! ( E_ LL\-cl a

+ Cr -otee)) d*

E#

Ftr...o( (h e

+GS*I

-r- ?

-

a

e-orea. I u.. 5 tc,( e {lne aryolJ

o o

)

Exz If-|

9qt s= 9oo

Ty':r+ cosgoV=tfro=l

@=?T ovsl+(os*

c,l A = rdrô( &

"t 0'= of = l+Cos o

f -- ttl=Lxg

0 =a

I

Y :o

I


Mathematics Lecture

2*class

ers tl(e5* "Tr\l- t (

&2-

2_o

t)ZF

sg?-

o

Cos& ++t(r+?rs

L

6 ??r

?irb

I o Z& cleo </\

e3c{ 9 1 Oiv'& + .L 5i ra?&g

o

3z

3I

a o o

1,0

I?

o@\J*

o

[+ I J,v

1,

t+ +case ++ + + coS zeJolo-

f3(T

Zrs


Mathematics Lecture

2"class

trxs ) fi.,.."/ the av. {-he ,eVda.^ iql lcl

X t+ t':

(x- t)"\-

th. ctrc (e +3' = | awJ o.lt5 r'c,{ c1'trc-Q

s ,/q.{r'o,.,r.

(x -r)

-'- Q= zcas *

x \ _:r

\t

q \

;7\ 2

-> xl:zx + I t3t= I

1-

Xt+Ut = z X ?-/ V = zFCoS&

t't t{ \

fl \r 1-

( = Lc-"5 cL

* A1 ,tr) ,qr.d

e C,S I- ?-

_T;--J3

il

rp

t3t = \

t

e


Mathematics Lecture

2*class

9

?

vz

e E3

f=ZG'S&L

x

Y

dfr :

e

V = ?ce,&

nol rob-? c?5O

1 ,lee

3

t fB

I

!-= I

zL

-T3 3

?

t (z c.5 *) -(t) lr I-,\

z

?4Cot e

"_lot o-

J

+3 7L-l

I-1\z cos& +l "la

t <{os z&)

J?\

E.) \

55/)

3

ztL

a ,/g

\


Mathematics Lecture

2*class

-La

3E

I-E

3

z L-L

"_66

?_3r

-5t

3

af

t

r + t + * s i,^(+ ))-(- !$ +Iri^ €lE +

? Jtt

rb


Mathematics Lecture

2*class

Ex{ I Fua +r- e 4\ *teT trctit^ fC\e) oVer (-tê re.frrbw R

srdea,ô,( o<rt5fo( e tk e <i y^cy'e vA-

Sr l uftb,-.. L

tt = I 9fz = l+CnS&Vt = V rl- tf CoJ&

-) CeS&=o

I

o

Z )

Y=1 elr ; l+Co;S&

x

-NL

g - _7\?-

e Ez f = l+ CoS&

Cv'tqdfr =

S= _?r

R

2.

lLl

ftbe) i"lrdaf = I


Mathematics Lecture

2"class

Fx5 : Evalu.{e Cxa'd'+t ) rlA utl,ev'e

R f'5 tkeR

f , tr'o \ i..,^ 5 tde *ke ctrc,/t X \3 t=4

.

|a-(,*tr'b*o ,. o<Y z ='{)

R

27f z

o o?TT

o z-Ti

o

(ô+J a+ Ddfr s (Y'+r )rolrclo

9=o f =o

X

9=o,- Xg =21'T

9=E:S f^L

e7t

(

o

z7T

o9- n=Z

?

Cr' +v) J tr-ob - _dfE: Ccyt{

o

K4" 4 lJo?_ ) -o

t?T

( rlrz) ole 6dct = 6elo

o

t,b: tz-ty


Mathematics Lecture

2*class

Ex 6:

(x +y w hre (ke Te f'ot,\l5

R

i ^ 1-h e €*'r, f(he clrc/rs

ucl r^<u",1,

<"'( bfiod qeu\f t b O U. r,..

X ?-t*3 1 aôl, X ?f ? 4

rr= I

x

V zCI+Y)dn= C fco5& +y'S1'vr&J y oJ f0@

T- O=b rf = t

?- ?_

e=*

R

Y'C Coss+Si.^. ce)Jn orl *o

?

I Ccos& fsi v ..a-) "l

g

a\

/

o tb

(.t<t'ff p"/ar rbub /e t,fuffro [ , euqlqqft

t)3

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Mathematics Lecture

2*class

Ta

2 )

3o

u\& &]

7l-o )

tsi \.\ CcE3 O

L I Csi,,. 9r2 - cosgo) ( Si*n o -(o! o) l3

l

/Ll

3

fil

3

Cass + si^ 4 ,l s

aI

l (t -_ o - C=, )lt


Mathematics Lecture

2*class

EX 7,, Cor,r grbleC f Lre y.e el'o,^ R Shoc{ w be/*^ts,,JV r'<\^ tS e\^ c (tSa"l bX 1z+J" = \ IKat3t=c{ ) $=X a,,Â i-6.At3.

9 I Ka cI

X

tr;// ,'ra fhe nrijsr,.n g/r,,îts ,g l,un te7 rqlib^ lFtf

CXty) d n tfc V't cg

A E

t o (nf,'ot

Te Lr, y),1 ft -- frrt&) Y d r-d&

I

u9

9 rhq

E


Mathematics Lecture

2*class

EX I I Ftr,.c,( t4'e q"reet 4 the gr'ue n u"e7,'o \

R bJ c.t(cu0q,rt'iqf ff r olv'o{& whe tre R.

fte YeA('o*^ ouf5r'"( e {k, etrc.(e f=Zol ins r d< Ze u ^&.n^w i reql e I L: g Ce f( ee)

IS

OtA

,(,^ tr'So O\-\ i {u-r\ -le}) fi)t) rr5 }Z z

)

2crs Cze)

3

V' ,-l )fq- =

y"?=9 coS(ze),t &--.,

!û = ? CISP>

i,ecasGe)

4 = 9 c.sc "e) -> ly'= + ,";i1):r t.;.L(

3

;3t

t)f:3

f , Llfl/


Mathematics Lecture

2*class

g)

og 5e=o$S n

9rog

*=o lr^s Z

Z

o

&

rol rotgI tl

- z +5)r^(ze)-1et o

2 [4-03 z-22 ]

Zc>

3 -62

9 cos zc9 @ =o-t55

A I&=o

o -55 I

l[n ?

F z

z LCsi* (zx''ss'J - q x o',-r,

o-555

( o- o) l


Mathematics Lecture

2*class

€x z I Fr*"I *he d.req "P t[.,e x. p V"(f ,f < L.tre"tlo .^ i^>iale 1L'e,rh, "l t-0..' e ci.cy'e

Carolr'ool lf = l+ Cag & ewo{l':: Co5@- r

5,.4rft'o^ '

!r.: 1a C"Sd-

,(2

ca56t-

r1r, +-p V" (-+ o f {ê cfncfle tf = CagO- tS SuePt@*t at z

('.2,^ $ Ifrf

o o f:o

e

CoS rP *

a

e r Ca\(*

ov dno(& = \+1 J*

Nz ry

t *(rt cotz&) Jso

o

z--o

I rle r+ Si,ta{Ll

?YI 2-\

q 2-+ +u/: -.-l

)

1.

Q.cP (a

eces &


Mathematics Lecture

2*class

7Le t"P v" ry n{ tl,e carol<'rJ r's gryept oq{by n - /1 C-S& a.5 * g'es f nt,, oErcSo t(S areo\ I5

e=N f= ltcoS& T

rol rol& - +? (r+ cosu)t Jr"@

0=o f=o o

1S

J / +LcoSe+CoS e),le

ds

= i((+ {r z +2r/z))-G

(

2

o r( 3 + ?Cos& +

zCo Cee

L

Z

2

T

fC, * z rps * q*C, rcoszq)"1 ,Y

-LZo

I

-N

t3A +zsiw&+ S irn C?e(

clL

-

Zo o

o

S .n(I

3NLl

+

Z>

L(r\


Mathematics Lecture

2*class

zr

6x \o ? F-run"{ tAe a,rea. i"side (he Cqno,kba{3 t z-Si* & qôl oqfs iole- cfrc.(e v =

rn64

)

tr:3?zSi*&

a

x

6

Y \I

qB-qw

*-- tfio

Q = ?7-. -+f = 3+Z Sir'r ?lo=J, + 2(-r)=

Lb

I

Thus t h, a re^ It^V ê€,+tb

r,r i S )

Y,

35

?r - 3; 2- Sir,rrg-

a* &= o -+

lr = 3 + 2 Sl.a.o : 3 1 o, s

S=9o" -)

f = 3 +-a SiuJo= 3+Z= !


Mathematics Lecture

2*class

-\ \?-

oo= -E a 2\:-+-E

6 €fC t{v^=

63 + 2 Si--T!-

z

Z

2z

3+ 2 S )wr&

6

)z 5i^8 + qs,'u &

.ft2

Vt= 3 t zslq + \, | 9t:) '\E {\'z : 2-

Y^t = Y2

3 +Z S i* & - 2 -.>

a Sl*,rg = -\

A f a\,/ôl&

p? Jsl.2

?

t ( J "lo-z

ea

, :ttc6

9+ ...--Z I ole


Mathematics Lecture

2*class

6

t+ f 6 sirn& a-?- Sitn?cr.] C*LZ

t 91'.,".61_ + ?- o5zez

-7\7T>6

t+ + 6 si,ô -cosze] Je-75>6

Crg S r] -^2&z a

6

- (*ry - 6 c-s (E ) -! s,',-.- (- f;lLt g

lz stt{ E,{ lz

-tefN +_fi8 :lLt-tgT3L

a

z5

,1

z

1r

Documents

Dept. Dr. 2*class