This document provides formulas and theorems related to trigonometric functions, differentiation, integration, limits, continuity, derivatives, and exponential and logarithmic functions. Some key points include: 1) Trigonometric formulas for sine, cosine, tangent, cotangent, secant and cosecant functions and their sums and differences. 2) Differentiation formulas for common functions including polynomials, trigonometric, exponential, logarithmic and inverse trigonometric functions. 3) Basic integration formulas for polynomials, trigonometric, logarithmic and inverse trigonometric functions. 4) Theorems on limits, continuity, derivatives, mean value theorem, extreme value theorem and properties related to concav

1. Formulas and Theorems for Reference
I. Tbigonometric Formulas
l. s i n 2 d + c , c i s 2 d : 1
sec2d
l * c o t 2 0 : < : s c : 2 0
I
+ . s i n ( - d ) : - s i t t 0
t , r s ( - / / ) = t r 1 s l /
: - t a l l H
7.
8.
sin(A* B) : s i t r A c o s B * s i l B c o s A
: siri A cos B - siu B <:os,;l
9. cos(A+ B) - cos,4cosB - siuA siriB
10. cos(A- B) : cosA cosB + silrA sirrB
2sirrd t:osd
12. <'os20- coS2(i - siu20 : 2<'os2o - I - 1 - 2sin20
11.
15.
13. tand :
14. <:ol0 :
I
<.rft0
I
t a t t H
: sitt d
(:ost/
sirrd
1
(:OS I/
1
ri" 6i
-el
-01
16. cscd -
/ F
I t l
r(. cos[ ^ l
18. : C O S A
215

2. 216 Formulas and Theorems
II. DifferentiationFormulas
!(r") - trr:"-1
Q,:I'
]tra-fg'+gf'
- gJ'-,f g'
-
,l'
, I
,i;.[tyt.rt)
- l''(tt(.r))9'(.,')
d ,
.7,
(sttt rrJ .* ('oqI'
t J , .
dr. l('os./ J stll lr
{ 1a,,,t,:r) -
"11'2.,'
o.t
1(<,ot.r')- (,.(,2.r'
Q : T
rl ,
,7,
(sc'c:.r'J: sPl'.r tall 11
d ,
- (<:s<t.r,;- (ls(].]'(rot;.r
, t
fr("')-.''
,1
fr(u")
- o,'ltrc
,l ,, 1
' t l l l r i -
( l . t ' . f
d,^ I
- i A l ' C S l l L l ' l
- - :
t!.r' J1 - rz
1(Arcsi' r) :
oT
*(i)
I l 1 2

3. Formulas and Theorems 2I7
III. Integration Formulas
1.
,f"or:artC
2. [0,-trrlrl *('
.t "r
3. [,' ,t.,: r^x| ('
,I
4. In' a,,: lL , ,'
.l 111Q
5. In., a.r: .rhr.r' .r r ('
,l
f
6.
./
sirr.rd.r' - ( os.r'-t C
7 . / . , , . r ' d r : s i t r . i '| ( '
.t
8.
,f'r^rr
tl:r:hr sec,rl+ C or ln Jccrsrl+ C
f
9.
.l
cot .r tlt lrr sirr.,l * C
1 0 . [ , n r ' . , , 1 . , l r r1 s c r ' . i* I a r r . r f C
.J
1i.
.[r,rr,rdr:]nlcscr
cotr] +C
12.
|
,"r' r d,r - tan r: * C
13. /*". r tarr.r'dr - sr'<'.r| ('
.l
14.
l
n""'r dr :-cotr:*Cl
15. /.'r.''t.ot r r/l' : ,'sr'.r r C
.t
16. [ ,urr'r cl.r- larr.r - .r+ ('
J
tT. [ ---!! -:lArctan({)+c
. l o ' 1 t " a a /
f ) -
18
Jffi:Arcsin(i)-.

4. 2t8 Formulas and Theorems
IV. Formulas and Theorems
1. Lirnits ancl Clontinuitv
A furrctiorry:.f (r) is c'ontinuousa,t.r- c if:
i) l'(a) is clefirrecl(exists)
ii)
Jitl,/(.r')
exists.and
iii) hru .l(.r) : ./(rr)
Othelrvise..f is <lisr:ontinrrorrsat .r'- rr.
Tire liniit lirrr l(r ) exislsif anclorrh'il iroth corresporrciirrgone-si<le<llinritsexist a,ncla,r'e
etlrtrl tlrtrt is.
lrgr,,l'(.r): L .:..=
,lirn,
.l'(.r) - I' -
,lirl
./(.r)
2. Intemrccliatc- rahreTheroettt
A func'tionlt , .l(r) that is r'orrtinrrt.rrrsr-rrra t:krserlinten'a,l fo.b] takes on every value
bct'uveerr./(rr) arrd./(6).
Notc: If ,f is corrtiriuorlsorr lrr.lr] an<1.l'(a) ancl .l'(1r)difler in sigrr. then the ecluatiou
.l'(.,)- 0 has at leu,stottesoirttiotritr the opetritrterval(4.b).
3. Lirrritsof Ilatiorial Frui<'tiorrsas .r + +:r;
if the <legreeof ./(.r') < thc clcglee of rt(r')
, . 2 ) , .
l ' . x ; r t r r 1 , l , ' :l i t , r
'
, .
' , ' - { l
.r'+r. .1"' ] .)
//, '2. lirrr
',
: is irrlirriteil tlre,legleeol ./{.r')' tlrerlegleeof 17(r)
. ,- t r
9 . 1 /
lirrr
/('] -o
. r '+ i l / . t J
/'/,)
3. litl # it
. r ' + f - r / ( . uJ
Notc: The limit
, ,. .rr + 2ll'
r.xiulll)l(': nlil L )c
.r'++x. J'' - ai
fiuite if the rlegteeof ./(.r:)- the degreeof .q(.r)
u,ill be the rtrtio of the leaclingc'ciefficientof .f(r;) to.q(r).
'2.r2-iJ.r -2 2
r - x a l l r l ) l c : l l l l r : -
t(),r'- 5r2 5

5. Formulas and Theorems 2I9
4. Horizontal anclrt'rtir:alAs)'rnptotes
1 . A I i n e g - b i s n l u r r i z o n t a l a s v n i p t o t t ' < - r f t h e g r a p h o f q : . / ( . r ' ) i f e i t h e r l i r r r l ( . r ' ;= l ;
,,r
.Itlt_ .f(r) : b
2. A lirie .t - e is a vcrti<'alas)'rrrptotc of tlie graph of tt - .f(.r) if eitirel
.,.hr,
.l(.,,)= *rc ur.
,)
./(.r')- +x.
5. Avcragc trrrrlIrrstarrtilll(-ollsIlat<' of ('lrarrgt'
1. Avt'ragt'Ratc of ('lratrgc:If (.r'9.yrr)attri (.r'l.ql) irle lroitrtsorr the glairlt <ftq - .l'(t).
tltert tlte a,velirg()ritte of c'harrgeof il u-ith rerspectto .r'ovcl tlrc itrtclr-al lr'11..rt; is
l!_r1'_l!,,) lr !1, ly
.l'1 .l'9 .r'l ,r'() l.r
'
2. Ittstatrtnrit'orrsRatc o1 (1-l',ltrg,',I1 (,r'1y..r/9)is a lroirrt orr the gralrlr oI rl ,-,.l'(.r).tiurrr
the itrstautArreoLlsrate of chirrigt,ofi7 n'ith rt,spt,r'tto.r' at ,r'11is .f''(.r'1;).
6. Dcfirritiorr of t,lrcl)r.rir-ativt'
.f'(.,)-lll lEP,r' t'(,,) 11,1,
!y)--ll:'J
Tlrt' la,tt<'rclcfirritiotrol tlrt' <k'tir';rtivt.is tlrt' irrstarrtirlr('()usrirtt, of charrgt' of' .l (.r) u-itlr
resltec:tto .t at .r -. (t.
Georrletrit'alir'.thtr <lerir':rtiveo1a fittlt'ti9lt at a lr,iltt is tlrt'sl'1re,f t1e'tatrg<'tttlitrt' t,
tho graph of the firnc'tion at tltat lioirrt.
7.
'fhc
Nrrrrrlrcr(' :ls a lirrrit
2. lini(1 + rr); (
n - )
8. Roller'sTheorerrr
If .l'is c't-rntituu.rttson ln.0] arrrlciiff'elentiablt'on(a.b) srrt'hthat.l'(rr).., l'(1,).tht'n thcle'
is at leirstottettutttberc itr the opetrintelval (o.b) srrc'hthat.l/(r') - 0.
9. Nlcan Valuc Thcorcrrr
If / is cotrtitnrortsott ln.lil aucl cliffelentiableon (o.f). then there is at 1t:astout' nurrilrer
l / 1 . I t ^ r
l iti (n.b; .tttlr tlt;tt
'/ "'t
-J)l!l-
-
f'1, I
t t
-
t I
-(1. li'r (r + 1)"
n + + a f l /

6. 1i)
220 Formulas and Theorems
Extreme - Vaiue Tlieorem
If / is contirmouson a closeclinterval lo.l.,].then./(.r) has both a tnaxinrum aurl a
minirnumon la.b].
11. To firid the rnaximrrrnand nrirrinuruvaluesof a furrc'ti<)tt =,/(.r'). loc'ate
1. the point(s) r,r'hclc .f'(.r) c'harrges sign. To firrri the c'atrcliclatesfirst fincl lvhcre
,f
'(.r:) - 0 or is infinite rlr cltterstrot t:xist.
2. thc t:trrlpoittts. if :rtn'. ort tltt' rlotttaitr<lf ,/(.r').
Corrrpalc' thc frurctiorr va,luesat trll of thcsc points lir firrrl the tnaxiruuuls an(l ntirtitttttttts.
Let ./ lic'cliffclcntialrit'firr rr <.1'< 1.,tttt<ltorttintrotrsfor rr { .r <. lt.
l. If ,f''(.r)> 0 for ('v('l''.r'irr(rr.L).therr.f is itrct't'asingorr frr.1l].
2. If ./'(.r'){ 0 for evelv.r'irr (o.L). tht'tt.f is clt't'rt'asrtrgorr [4.1l].
Srippr-,seth:rt .f'"(;r) t'xists ort tlte itrtelva,l(rr.lr).
1. If ,f"(t') ) 0 irr (a.b).tlrcn.f is <'orrcr,veupu,rrr'<lirr (a./r).
'). If .f"(.r) { 0 irr (rr.L).tlrerr.f is corrc'tr,ve(lo$:lrwfrlclirr (rr./r).
To lot'trtethe pointsof irrfkrc'tir.rrttfi tt -.1'(.r').firxl the proitrtsr'vhere.l'"(r')- () or u'ltt'r't:.f"(.r')
fails to cxist. l'irest,'arethe orrh'r'uclirl'r1,'t;lyllere.f(.r')rnar.hal't'a poirrtof irillectitxt.
'Ilten
test tlresepoints to urirkcsuretha,t,l'"(.,).- 0 on ont'sitlt'arrtl ,f"(.r) > 0 <.rtttlu'other'.
1.1 Diffcrerrtialrrlitv irnplies r'ontiuuitt': If a frrnr:tiorris cliflereltialrlt' a,t a poirrt .r'- rr. it is
t'<.irrtinuousat that 1.loirrt.
'I'he
convcrst'is falst'.i.e. c'ontintritvrkrcsnot iurpll'cliffert'ntiabilitr..
15 LorrtrlLirr<'aritr-arr<1Litrcal Approxittratiorr
'l'iie
liriear trpproxitnzrtiottof ./(.r')rrear.t'-.t0 is giverrlx'4:./(.,'e) *.1'(.l'1)(.r' .re).
Tir estiuratc the slope of a gralrh at a poirrt rha,n a trrngerrt lirx-'to tltc graph at tliat point.
Arrother rva. is (lx' using u grtrphit s cak'nla,tor') to "zoonr in" aroLtn<lthe point itt cluestiorr
urrtil the glaph "kroks'' straight.'fhis rrretliocl alnrost ahva'"s il)r'ks. If u'c' "zot.rtttin" att<l ther
glaph Lr,rks stlaiglrt at a point. sa)'.r': o. then the funr:tiorris loca,ll)'lincar at that point.
flre graph of u : ].r:lhas a sharp (:olner' .rt :f :0. This col'll€rrc'arlllot lre stlrot-rtheclout lte
"zc.ronringin" r'epeatecllv.Consecluetrtll'.the clerivative of l.r' cioesnot exist at .r': 0. henc'e.is
not locallr' Iinear at .r': 0.
12
l ' )
_ t , ) .
l

7. Formulas and Theorems 221
Tlrt't'xpotretrti:rl func'tir)u!: c' gt'<lu'sverv lapirlh.AS.r'-+ tc u,h.ilethe fttgarithmic,fulr.tion
l/ .. lrr.r' glo's vt'r'r' skx.r,i-u'a.s.r'-) )c.
Erpotrerttial frruc'tiorrslike u -. 2' rtr !/ : r,,'llr.()-ntol.er:rpiclly as.r +:r tharr an), positive
l)()'('1<if .r.
'1.'ht'fitttt'tiott
i/ - hr.r' gr'o'ssl<lu'eras .t -+ x tltiil a1r lotx,orrstarrt lt1;lvrr<1niai.
i' sar'. that as .r'-+ )c:
l . I t . tt g l ' ) ' :l ; r - 11 , 1 .l l r i r r r, / i , rI i l l i r r r
l ( r - , r ' i l l i r r r
l t | ' )
{ t .
r . r z / { , r ' ) . r . . l ( . r ' )
fi.l (r') gltxls fhster thatr a(.r')as.r'-+ )c. therrq(,r')gr'owsslolr,tr tlu.rn.l'(.r.)AS.r. + rc.
2. ./(.r) arr<lr7(.r')grou, at the sarnt'ratt,as .r' + r if lir,r
'19
L l0 (tr is firrite ancl
, . , q ( . r , )
rrouzt'r'o).
Fol t'xanrlllt'.
1. r' gtrxls l;rstcr tlrarr.r.:iils.r, + rc sirrr.r,lirrr { -. :r,
. t '
'2.
.r'l gr',,1'slirstcl tlrarr hr.r' :rs .r. : rc sirr<.e1i,,,-
,'1
x
3. .r': + 2.r'gl()'sirt tll, sirlrrt'r'rrtr',rs .,,1as .r. )
'r2 l2
>csirr<.r'
,]11 ,i{
I
Tir firl<lsotttt'of the'st'litrritsirs ,r' , . '()llnrirv lrs('the graphing talr.rrlat.,r'.Iake su.c,tlat
alr al)l)l()l)liatc r.it'u-irrgr.l-itrrlori-is rrscrl.
17. Irrr-t'r'scFrrru'tiorrs
1Li. CourlraringRatcsof C'hatrgc
i. If ./ lrrrl 17irlt' tu,o frrrr<.tioussrr<.hthat .l'(q(.r.))- .r for e-,()1...1,in tiu, rlorrrairrol q.
arrtL.q(.1'(.r')) .r'.lirr irr thc'rlolrairr of .f. therr. .f' arrd 17are irrvelst'fiurr.tions
til eirchotlrcr.
'2.
A ftlrrr'1iorr.f htls rttt itrvt'rsr'lirttttiou if arrrl onh. if rio lrorizorrtal liue irrtcrserr,tsits
gralrlr urolt' tlrirrrorr<'(r.
3. If .l is t'itlrt't ittt t'eilsilg or' <it'treasirrg in arr intt:r'val. tfien f' |as a1 i1.,r'r.sefilrc:ti,'
or.t't thrrt irrtt'r't'al.
l. h I is tlilfi'rt'rrtia]rlt' at t'vt't'r-lroirrt ori arr irrterval I. arrcl ,f'(.,t)I0 orr I. t1e1
!l '- l
r(.,
I is tlifTt'r<'utitrlrlt'at everr'lroint of the interior of the interval l'(I) arrrl
, t ' l l l . rI )
l
|
'
r. t t
'

8. 222 Formulas and Theorems
I x P r r r rr l t t i l s r , 1 r ' '
' _ - l - ' - _
1' I'htr t'xllorlt'utial futtctit.rti !/ - t'' is the irlverse function of t7:111 .1''
2. I'lrt'clornaitt is thc set rlf all rt'al rltlrlll)el's.-)c <.lr < DC.
3.
'l'lu'ritngt'is
tlrt'set of all llositive nttntllels.! > 0.
, l
- 1 . - l ( ,
' ) , '
.
( L t '
5. ll .,r' is <'ontirtlrorrs.inc'r'errsirrg.attd (on('irve rtlt fbl all .r:.
t t .
,
i i t ] ' _ , ' . , i x a t r t l
, .
l 1 t l t _r ' - ' 0 .
T . , l t tr . r . .f i r r '. r .- >0 l l r r ( r ' ) - . r ' f i r r a l l . r ' .
19. Prolrt'r'tit's o[ ]tt.r'
1.
'l'lrc
rkrrrririuo1 r7 lrr,r'is tht: settof all ltositivc trutttliers,.r'> 0.
'2. '['lrt'rirrrgt'of
i7 . hr.r' is tlie sct of all rt'al lrtrttt]rers. x < l/ < :r'
:1. r7 . lrr.r' is <.orrlirrrrorts.itr<'r'e'asirrg.urrrl corrcavtrclou,ttcverYrvltertl r-rttits tlclrltrin.
1 . l r r ( r r | )- l t r r r I l t r1 i .
1 . l t r l , tf l , I l t r , r l r r/ ,
( ; . 1 1 11 1 l , . . 1 ' 1 1 v1 1
7. i7 hr.r '- 0 iI 0 .: .r'.- I arrrllrr.r'> 0 if .r > 1.
E.
,lllt.
ltr.r'- *:r trrtrl
,.lt]li
ltt.r'- -)c'
1).l.g,,.r'il;
20. 1-tlpczoitlirl Ilrrlt'
If ir furt.tiorr.fis c'outiuuorrsorrtlrt't'krseclinte't'val[4.b]where fo.b]has ]reenpartitioned
irrtrr l sttlrirttt,r'r'trlsI.r'1..r'rj.l,r.i2].....[.r:,, r..t:,,].ent:ltof length (b-a)ln. then
r l t r
I f t,) r/.r'= -;;[./(,0) +'2.f(.rr)+ 2/(.rz)+ ... + 2J(.r',,r) + ./(.r")].
. t, ,
Tlrt. T'ralrezoiclal Rrrlt' is tlre avelage of the left-hancl and riglrt-hancl R,iemann sulns.

9. Formulas and Theorems 223
Let ,/(.r) and r7(,r) be c,cintirruuousorr la. ll].
fb rt,
1.
J,,,,'.f
(r) rl,t': c,l,,.rrr,r1.r.r'is a uor.zcroc,onstant.
f t
2'
.1,,
f ('') rl'rr- 0
I ' t t | t '
:l
.1,,,,')'ltr
-
.f,,
lt,t,t,
[t ' r ' lt'
+.
.1,,,r,),lr- f,,.1t.,)n.,*,1,.f'(.r)rl.r..
r,r.her.t',f is continrrouson arr irrter.val
r'orttailrittgtlte trutnltet'srr. 1r.arrrlr'. r'egarrllt'ssol tlrt'or'<lt'r.a.|. arrclr,.
5. If l(.r')is trn otlrlfiurr.tion.th,',r / .l(r') rl.t.- (l
.l ,,
tj. If ./(.r) is arrevenfiul.tion. tlruu
.f
",,.1'(.t.)
tlr I
,f,,'
.l{.,) ,t,,.
21. Propcrtiesof tlic Dcfinitc Ilttcgral
22. Dcfiriti.' .f D<'firritt'hrt<'gr:rltrs tli. Li'rit ,f u Srrrrr
Sttlllrtlst' that a firtttrtiott .l'(.r')is <ontirurorrsorr the r.krst'<lirrtelval lrr.li]. Divirle t1e
iritt'rval irrto rr cclrral sulrirrtcrvals.ol length A.r.
l:
" (,h,,,,st,(,ll(, nl1lnl)er.irr caclr
sttbitrtt'lval i.t'. .r'1in tlu' filst. .r'2irr tlrt, st,r'orrrl.. . . ..r'A.'i1tl,,' A.th.. . .. ^rrrl .r.,,i, tlr. rrt5.
'
r l ,
'Ilicu,,lirrr
I
r, r'1)Jr'-
f,,.f
{.,),t.,
2:1. Funrlarncntal
'flrlollrrr
,1 ('ak.uhrs
7 b
I tt.,) ,l.t l:iltt 1-'trit. n"lu,r.t,F,(.r) : ,f(..r')
.t,,
7 II .l(.r)] 0 on lrr.1r].rherrl'",,(,,,r/.r,> 0
.t,,
8. If .q(.r')Z.f(r),n lo.bl.rl*,u
[,,",,{.r),1,r
7
,[,,".1{.,.1
,t.,
o,+..f',,',,,,,,, ri, ,',',rj,
f,"''',,rr,
rtt:,f(q(t.))g,(.r).

10. Formulas and Theorems
24. Y"t".lty, Sp..a,
"t
1. The vclocity of an object tells how fast it is going and in which direction. Velocity is
an instantaneous rate of change.
2. The spceclof an obiect is the absolute value of the velocity, lr(t)I. It tells how fast
it is going disregardingits direction.
The speeclof a particle irrcrcascs(speedsup) when the velocity and accelerationhave
thersarrresigns.The speedclecreascs(slows down) when the velocity and acceleration
have oppositesigns.
3. The acr:cier:rtionis thc irrstantarreousrate of change of velocity it is the derivative
c-rfthe veloc:ity that is. o(l) : r"(t). Negativeacceleration(deceleration)meansthat
t[e vgloc:ity is dec:r'easirrg.Tlie acceleration gives the rate at which the velocity is
crharrging.
Therefore,if .r is the displacernentof a rnoving objec:tand I is time, then:
i) veloc:itY: u(r) : tr (t :
#
ii) ac'creleration: o(t): ."'(t): r'/(/)-
#.
:
#
iii)i'(/) [n(t1,tt
iv) .r(t)- [ ,,31a,
Notc: T[e av('ragcvelclcity of a partir:le over the tirne interval frorn ts to another time f. is
Averagevel;c'itv:
T#*frH#:
"(r] -;'itol. wheres(t) is thep.sitionof
the partic:leat tinre t.
25. The avetage value of /(r) on [a. ir] is f (r) d:r.
Arca BctwtxrriCtrrvt,s
If ./ anclg are continuousfuncrtionssuchthat /(:r) 2 s@) on [a,b],then the area between
,.b
I
I lrecrrrvesis / l/ (",I - q(rl) dr .
J a
+,,,,1,,'
26

11. Formulas and Theorems 225
27.
28
Volume of Soiids of R.evolution
Let / be nonnegativeand continuouson [a,.b]. and let R be the region bounded above
by g: /(r"). belowby the r-axis, and on the sidesby the linesr:: n and r:b.
When this region .R is revolved about tire .r'-axis.it gerreratesa solid (having circular
f o
crrosssec'tions)u'hosevolume V -
| {j'(.,'l)2 ,1.,.
/ t t
Volunrcsof Soli<lswith Knowrr CrossScctions
l . of area A(:r:). taken pt'r'lierrcli<'ulartcl the r-zrxis.
d r .
A(37)taken perpt'rrriicrrlarto the 37-axis,
29. SolvirrgDifferential Equations:Graphically ancl Nurnerrir.all.l'
SkrpcFieicls
Af ever'1'poirrt (.r.r7)a differetrtialecluatiorrof the folrrr #
- f t, .i/) gives the slope of tht'
nernber of the farnily of solutit.rnsthat c:onta,insthat poirrt. A slope fielcl is a, gra,lrhictrl
represent:rtiotrof this family of curves.At eac:hpt-rirrtirr the plarre.a short s()gnlentis rlrau'n
"vhose
slopeis eclualto the value of the clerivativerat that poirrt. I'hesescgnrerrtsare taugcnt
to the sohrtion'sgraph at the poirrt.
The slope fielcl allows you to sketc:hthe graph of thersolution cul've even though you rlo rrot
have its ec|ration.This is clc-rneby starting at arrypoint (usuallv the point given bv the initial
c'ondititin).and moving fron one poirrt to the next in the direc'tionirrdicntedby the segrncnts
of the slope fielcl.
Somct'trlc'ulatorshavttbuilt in operationsfbr drawing slopefields;fcircalculatorsrvithorrttiris
feature tlrere are l)rograms available fbr drawing thern.
30. Soiving Diffelential Equations b)' Separatirrgthe Variables
There are lnAny technicluesfor solving differential equations. Any differential equatir_rnvou
may be askedto solveott the AB Calculus Exam can be solvedby separatingthe variables.
R,ewritethe equatioll as an erluivalent equation with all the r and dr terrns on otle side arxl
all the q and d37terrns ou the c-rther.Antidifferentiate both sides to obtain an e(luation
without dr or du, but with ortec'onstantof inteqration. Use the initial condition to evahrate
this constant.
of area
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Fol crosssections
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