Formulas and Theorems for Reference
I. Tbigonometric Formulas
l.
sin2d+c,cis2d:1
sec2d
l*cot20:<:sc:20
I
+.
sin(-d) :
-sitt0
t,rs(-//) = t r1sl/
-tallH
:
7.
s i n ( A* B ) : s i t r A c o s B * s i l B c o s A
8.
: siri A cos B - siu B <:os,;l
9.
cos(A + B) - cos,4cosB - siu A siriB
10. cos(A - B) : cosA cosB + silrA sirrB
2 sirrd t:osd
11.
12.
< ' o s 2 0- c o S 2( i - s i u 2 0 : 2 < ' o so2 - I - 1 - 2 s i n 2 0
I
< . r f0t
13. tan d :
14. <:ol0 :
I
tattH
sirr d
1
15.
(:OS I/
1 6 . c s cd -
/
r(.
(:ost/
F
I
tl
cos[ ^
1
ri" 6i
-el
:
sitt d
-01
:
COSA
\l
18.
215
Formulas and Theorems
216
II. Differentiation Formulas
!Q,:I'
(r")
- trr:"-1
]tra-fg'+gf'
* (i)
-- g J ' - , f g '
,l'
,I
( t t ( .)r) 9 ' ( . , ' )
,i;.[tyt.rt) l''
d,
\
(sttt rrJ .* ('oqI'
.7,
.
stll lr
tJ,
\
./
dr. l('os J
:r) - "11'2.,'
{ 1a,,,t,
o.t
.r')1(<,ot
Q:T
(,.(,2
.r'
rl ,
,7,
: sPl'.r tall 11
(sc'c:.r'J
d,
.r,; - (<:s<t
(ls(].]'(rot;.r
-.''
fr("')
,t
,1
fr(u") o,'ltrc
,l ,,
'
(l.t'
tlll ri -
d,^
-iAl'CSllLl'l
1
.f
--:
I
t!.r'
J1 - rz
r) :
1(Arcsi'
oT
Il12
Formulas and Theorems
III. Integration Formulas
1.
,f"or:artC
*('
2.
[\0,-trrlrl
.t "r
3.
[ , ' , t . ,: r ^ x| ( '
,I
4. In' a,,: lL
, ,'
111Q
.l
5.
I n . , a . r : . rh r . r ' . r r ( '
,l
f
6.
sirr.rd.r'
- ( o s . r '- t C
./
7.
8.
/ . , , . r ' d r : s i t r . i '| ( '
.t
,f'r^rr
f
9.
t l : r: h r s e c , r+l C o r
cot .r tlt
.l
10.
1i.
12.
13.
14.
15.
[,nr'., ,1.,
.J
l r r s i r r . ,l * C
l r r1 s c r ' . *i I a r r . r f C
cotr] +C
.[r,rr,rdr:]nlcscr
,"r' r d,r - tan r: * C
|
/ * " . r t a r r . rd' r - s r ' < ' . |r ( '
.l
n " " 'r d r : - c o t r : * C l
l
/.'r.''t.ot r r/l' :
.t
,'sr'.r r C
1 6 . [ , u r r 'r c l .-r l a r r. r - . r + ( '
J
tT.
18
[ ---!! -:lArctan({)+c
.l o'1t"
f
ln Jccrs+
rl C
a
\a/
)-
Jffi:Arcsin(i)-.
2I7
2t8
Formulas and Theorems
IV. Formulas and Theorems
1.
Lirnits ancl Clontinuitv
A f u r r c t i o ryr : . f
i)
ii)
( r ) i s c ' o n t i n u o uas, t . r - c i f :
l'(a) is clefirrecl(exists)
Jitl,/(.r')
e x i s t s .a n d
i i i ) h r u . l ( . r) : . / ( r r )
at .r'- rr.
Othelrvise..f is <lisr:ontinrrorrs
linrits exist a,ncla,r'e
Tire liniit lirrr l(r ) exisls if anclorrh'il iroth corresporrciirrg
one-si<le<l
etlrtrl
tlrtrt is.
lrgr,,l'(.r): L .:..=
2.
,lirn,
.l'(.r) - I' -
,lirl
./(.r)
Intemrccliatc- \rahre Theroettt
r-rrra t:krserlinten'a,l fo. b] takes on every value
A func'tion lt , .l (r) that is r'orrtinrrt.rrrs
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,stotte soirttiotritr the opetr itrterval (4.b).
3.
Lirrritsof Ilatiorial Frui<'tiorrs
as .r + +:r;
lirrr
. r '+ i \
/('] -o
l/\.t J
if the <legreeof ./(.r') < thc clcglee of rt(r')
l ' . x ; r t r r 1 , l ,l' i:t , r
.r'+r.
'2.
lirrr
. ,- t r
.r'+f -
]
.)
9\.1/
,
litl
.1"'
'/ ,/ , \
i l t l r e , l e g l e eo l . / { . r ' )' t l r e r l e g l e eo f 1 7r()
: is irrlirrite
r.xiulll)l(':
3.
', . 2 ') , ', . {l
,.
,.
nlil
.r'++x.
.rr + 2ll'
L
J'' -
)c
ai
/ ' / ,)
it fiuite if the rlegteeof ./(.r:)- the degreeof .q(.r)
#
r/(.uJ
Notc: The limit u,ill be the rtrtio of the leaclingc'ciefficient
of .f(r;) to.q(r).
' 2 . r 2- i J . r - 2
r-xallrl)lc: llllr
t ( ) , r-' 5 r 2
:
-
2
5
Formulas and Theorems
4.
Horizontal ancl \rt'rtir:al As)'rnptotes
1. AIineg-bisnlurrizontalasvniptott'<-rfthegraphof
,,r
2.
q : . / ( . r ' ) i f e i t h e r l i r r r l ( . r ' ;= l ;
(r) : b
.Itlt_ .f
A
lirie
.,.hr,
5.
2I9
.t - e
is
. l ( . , , )= * r c
a
ur.
vcrti<'alas)'rrrptotc of
tlie
graph
of
tt - .f(.r)
if
eitirel
. / ( . r ' )- + x .
,\)
A v c r a g c t r r r r lI r r s t a r r t i l l l ( - o l l sI l a t < ' o f ( ' l r a r r g t '
1 . A v t ' r a g t ' R a t c o f ( ' l r a t r g c : I f ( . r ' 9 . y r ra) t t r i ( . r ' l . q l ) i r l e l r o i t r t so r r t h e g l a i r l t < f t q - . l ' ( t ) .
t l t e r t t l t e a , v e l i r g ( r) i t t e o f c ' h a r r g eo f i l u - i t h r e r s p e c tt o . r ' o v c l t l r c i t r t c l r - a l l r ' 1 1 . .t r; i s
l!_r1'_l!,,)
.l'1
2.
6.
.l'9
lr
.r'l
!1,
ly
,r'()
l.r
'
, ( , r ' 1 y . . r /i 9
I t t s t a t r t n r i t ' o r rRsa t c o 1 ( 1 - l ' , l t r g ,I' 1
s )a l r o i r r t o r r t h e g r a l r l r o I r l , - , . l ' ( . r ) .t i u r r r
t h e i t r s t a u t A r r e o L rl sa t e o f c h i r r i g t , o f i 7 n ' i t h r t , s p t , r ' t o . r ' a t , r ' 1i1s . f ' ' ( . r ' 1 ; ) .
Dcfirritiorr of t,lrc l)r.rir-ativt'
.f'(.,)-lll lEP,r'
!y)--ll:'J
t'(,,) 11,1,
T l r t ' l a , t t < ' rc l c f i r r i t i o t ro l t l r t ' < k ' t i r ' ; r t i v t .i s t l r t ' i r r s t a r r t i r l r ( ' ( ) u rsi r t t , o f c h a r r g t ' o f ' . l ( . r ) u - i t l r
resltec:t to .t at .r -. (t.
G e o r r l e t r i t ' a l i r ' .t h t r < l e r i r ' : r t i v e o 1a f i t t l t ' t i 9 l t a t a l r , i l t t i s t l r t ' s l ' 1 r e , f
tho graph of the firnc'tion at tltat lioirrt.
7.
'fhc
t1e'tatrg<'tttlitrt' t,
N r r r r r l r c r( ' : l s a l i r r r i t
1.
li'r (r + 1)" -(
fl /
\
n++a
2 . l i n i ( 1 + r r) ;
(
n -\)
8.
Roller'sTheorerrr
If .l'is c't-rntituu.rtts
on ln.0] arrrl ciiff'elentiablt'on(a.b) srrt'hthat.l'(rr).., l'(1,).tht'n thcle'
is at leirst otte ttutttberc itr the opetr intelval (o.b) srrc'hthat.l/(r') - 0.
9.
Nlcan Valuc Thcorcrrr
I f / i s c o t r t i t n r o r t so t t l n . l i l a u c l c l i f f e l e n t i a b l e o n ( o . f ) . t h e n t h e r e i s a t 1 t : a s to u t ' n u r r i l r e r
l / 1 . \ - J )I lt !^lr'/
l iti (n.b; .tttlr tlt;tt "'t f'1, I
tt
tI
220
Formulas and Theorems
1i) Extreme - Vaiue Tlieorem
If / is contirmouson a closeclinterval lo.l.,].then./(.r) has both a tnaxinrum aurl a
m i n i r n u mo n l a . b ] .
11. To firid the rnaximrrrnand nrirrinuru valuesof 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
'(.r:) 0 or is infinite rlr cltterstrot t:xist.
,f
2.
thc t:trrlpoittts. if :rtn'. ort tltt' rlotttaitr <lf ,/(.r').
Corrrpalc' thc frurctiorr va,lues at trll of thcsc points lir firrrl the tnaxiruuuls an(l ntirtitttttttts.
12
l')
_t,).
L e t . / l i c ' c l i f f c l c n t i a l r i t ' f i r r r r < . 1 ' < 1 . t, t t t < lt o r t t i n t r o t r s f o r r r { . r < . l t .
l.
I f , f ' ' ( . r )> 0 f o r ( ' v ( ' l ' \ ' . r ' i r r( r r . L ) . t h e r r . f i s i t r c t ' t ' a s i n go r r f r r . 1 l ] .
2.
I f . / ' ( . r ' ){ 0 f o r e v e l v . r ' i r r ( o . L ) . t h t ' t t . f i s c l t ' t ' r t ' a s r t rogr r [ 4 . 1 l ] .
th:rt .f'"(;r) t'xists ort tlte itrtelva,l(rr.lr).
Srippr-,se
1 . I f , f " ( t ' ) ) 0 i r r ( a . b ) . t l r c n . f i s < ' o r r c r , vuep u , r r r ' i<r lr ( a . / r ) .
'). If
(lo$:lrwfrlcl
irr (rr./r).
.f"(.r) { 0 irr (rr.L).tlrerr.f is corrc'tr,ve
tfi tt -.1'(.r').firxl the proitrtsr'vhere.l'"(r') - () or u'ltt'r't:.f"(.r')
To lot'trtethe points of irrfkrc'tir.rrt
'Ilten
lyllere .f (.r')rnar.hal't'a poirrt of irillectitxt.
fails to cxist. l'irest,'arethe orrh'r'uclirl'r1,'t;
tlu'other'.
test tlresepoints to urirkcsure tha,t ,l'"(.,).- 0 on ont'sitlt'arrtl ,f"(.r) > 0 <.rtt
1.1
Diffcrerrtialrrlitv irnplies r'ontiuuitt': If a frrnr:tiorris cliflereltialrlt' a,t a poirrt .r'- rr. it is
'I'he
convcrst'is falst'. i.e. c'ontintritvrkrcs not iurpll'cliffert'ntiabilitr..
t'<.irrtinuous
at that 1.loirrt.
1 5 L o r r t r lL i r r < ' a r i t r -a r r < 1L i t r c a l A p p r o x i t t r a t i o r r
'l'iie
l i r i e a r t r p p r o x i t n z r t i o tot f . / ( . r ' )r r e a r . t ' - . t 0
i s g i v e r rl x ' 4 : . / ( . , ' e )
*.1'(.l'1)(.r' .re).
rha,n a trrngerrt lirx-'to tltc graph at tliat point.
Tir estiuratc the slope of a gralrh at a poirrt
(lx'
Arrother rva\. is
using u grtrphit s cak'nla,tor') to "zoonr in" aroLtn<l the 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:l has a sharp (:olner' .rt :f :0. This col'll€rr c'arlllot lre stlrot-rtheclout lte
"zc.ronringin" r'epeatecllv.Consecluetrtll'. the clerivative of l.r' cioes not exist at .r' : 0. henc'e. is
not locallr' Iinear at .r' : 0.
l
Formulas and Theorems
221
1Li. CourlraringRatcs of C'hatrgc
T l r t ' t ' x p o t r e t r t i : r l f u n c ' t i r ) u! :
c ' g t ' < l u ' sv e r v l a p i r l h . A S . r ' - + t c u , h . i l et h e f t t g a r i t h m i c , f u l r . t i o n
..
lrr.r'
glo\\'s
vt'r'r'
skx.r,i-u'
a.s .r' -) )c.
l/
E r p o t r e r t t i a l f r r u c ' t i o r r sl i k e u - . 2 ' r t r ! / : r , , ' l l r . ( ) \ \ - n t o l . er : r p i c l l y a s . r + : r
tharr an), positive
'1.'ht'fitttt'tiott
l)()\\'('1<if .r.
i/ - hr.r' gr'o\\'s sl<lu'eras .t -+ x tltiil a1\r lotx,orrstarrt lt1;lvrr<1niai.
\ \ i ' s a r ' . t h a t a s . r '- + ) c :
l(r\
I t . tt g l ' ) \ \ ' : l ; r - 1 , 1l .l r i r r r, / i , rI i l l i r r r
l.
r
.r
z/{,r')
- \
,r'il
lirrr
lt|')
{t.
.r .\.l(.r')
f i . l ( r ' ) g l t x l s f h s t e r t h a t r a ( . r ' ) a s . r ' - + ) c . t h e r r q ( , r ' ) g r ' o w ss l o l r , t r t l u . r n . l ' ( . r . )A S . r . + r c .
2.
'19
if lir,r
,. ,\ q(.r,)
. / ( . r ) a r r < lr 7 ( . r ' )g r o u , a t t h e s a r n t ' r a t t , a s . r ' + r
L l0
(tr is firrite ancl
rrouzt'r'o).
Fol t'xanrlllt'.
1 . r ' g t r x l s l ; r s t c r t l r a r r . r . : i l s . r , + r c s i r r r . r ,l i r r r
{. t
'2.
3.
'
-. :r,
. r ' l g r ' , , 1 ' sl i r s t c l t l r a r r h r . r ' : r s . r . : r c s i r r < . e1 i , , , ,'1
. r ' : + 2 . r 'g l ( ) \ \ ' si r t t l l , s i r l r r t 'r ' r r t r ,' r s . , , 1a s . r . )
x
>c sirr<.r'
,]11
'r2 l2
,i{
I
T i r f i r l < ls o t t t t ' o f t h e ' s t ' l i t r r i t si r s , r ' , \ . \ ' ( ) l l n r i r v l r s ( ' t h e g r a p h i n g t a l r . r r l a t . , r '\ .I a k e s u . c , t l a t
a l r a l ) l ) l ( ) l ) l i a t c r . i t ' u - i r r gr . l - i t r r l o riis- r r s c r l .
1 7.
I r r r - t ' r ' sF
c r r r u ' ti o r r s
i.
'2.
I f . / l r r r l 1 7i r l t ' t u , o f r r r r < . t i o u ss r r < . ht h a t . l ' ( q ( . r . ) )- . r f o r e - , \ ( ) 1..1\ .,i n t i u , r l o r r r a i r ro l q .
a r r t L .q ( . 1 ' ( . r ' ) ) . r ' . l i r r
i r r t h c ' r l o l r a i r r o f . f . t h e r r . . f ' a r r d 1 7a r e i r r v e l s t ' f i u r r . t i o n s
til eirchotlrcr.
A f t l r r r ' 1 i o r r. f h t l s r t t t i t r v t ' r s r ' l i r t t t t i o u i f a r r r l o n h . i f r i o l r o r i z o r r t a l l i u e i r r t c r s e r r , t ist s
g r a l r l r u r o l t ' t l r i r r ro r r < ' ( 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 i s t l i l f i ' r t ' r r t i a ] r l t ' a t t ' v t ' t ' r - l r o i r r t o r i a r r i r r t e r v a l I . a r r c l , f ' ( . , t )I 0
orr I. t1e1
r(., is tlifTt'r<'utitrlrlt'
'
I
l
at everr'lroint of the interior of the interval l'(I) arrrl
!l
l
,t'll l.rI)
'
'
r
| .tt
222
Ix
Formulas and Theorems
P
r,1 r ''
' _ r
- l r r rr -l ' t- _t i l s
1'
.1''
I'htr t'xllorlt'utial futtctit.rti !/ - t'' is the irlverse function of t7:111
2.
I ' l r t ' c l o r n a i t t i s t h c s e t r l f a l l r t ' a l r l t l r l l l ) e l ' s .- ) c
'l'lu'ritngt'is
3.
,l
-1. -l(,
(Lt'
tt.
,
i s < ' o n t i r t l r o r r si .n c ' r ' e r r s i r r ga.t t d ( o n ( ' i r v e r t l t f b l a l l . r : .
iit]'_,'
r
T. ,ltt
tlrt'set of all llositive nttntllels.! > 0.
, ' .
')
ll .,r'
5.
<.lr < DC.
., i x
a t r t l l 1 t l t _r ' - ' 0 .
,.
. r . .f i r r '. r .- > 0 l l r r ( r ' ) - . r ' f i r r a l l . r ' .
1 9 . P r o l r t ' r 'itt ' s o [ ] t t . r '
1.
'l'lrc
r k r r r r i r i uo 1 r 7
l r r , r ' i s t h t : s e t to f a l l l t o s i t i v c t r u t t t l i e r s , . r ' > 0 .
'2. '['lrt'rirrrgt'of i7 . hr.r' is tlie sct of all rt'al lrtrttt]rers. x
< l/ < :r'
urrrl corrcavtr clou,tt cverYrvltertl r-rttits tlclrltrin.
:1. r7 . lrr.r' is <.orrlirrrrorts.itr<'r'e'asirrg.
1 . l r r ( r r | )- l t r r r I l t r 1 i .
1.
l t r l , ft l , I
(;.
1 1 1 1 l , . . 1 ' 1 11v 1
7. i7
E.
ltr,r
l r r/ ,
hr.r '- 0 iI 0 .: .r'.- I arrrllrr.r'> 0 if .r > 1.
,lllt.
ltr.r'- *:r
ltt.r'- -)c'
trrtrl
,.lt]li
1).l.g,,.r'il;
20.
1 - tl p c z o i t l i r l I l r r l t '
orr tlrt't'krseclinte't'val[4.b] where fo.b] has ]reenpartitioned
If ir f\urt.tiorr.fis c'outiuuorrs
i r r t r r l s t t l r i r t t t , r ' r ' tIr. lrs' 1 . . r ' rl j, .r . i 2 ] . . . . . [ . r : , , r . . t : , , ] .e n t : l t o f l e n g t h ( b - a ) l n . t h e n
rlt
r
.
I f t , ) r / . r=' - ; ; [ . / ( , 0 ) + ' 2 . (f . r r )+ 2 / ( . r z )+ . . . + 2 J ( . r ' ,r,) + . / ( . r " ) ]
.t
,,
Tlrt. T'ralrezoiclal Rrrlt' is tlre avelage of the left-hancl and riglrt-hancl R,iemann sulns.
Formulas and Theorems
223
21. Propcrties of tlic Dcfinitc Ilttcgral
Let ,/(.r) and r7(,r) be c,cintirruuousorr
la. ll].
fb
rt,
( r ) r l , t ' : c , l , , . r r r ,r 1 . rr.' i s a u o r . z c rco, o n s t a n t .
J,,,,'.f
1.
ft
2'
f ('') rl'rr- 0
.1,,
:l
.1,,,,')'ltr
I'tt
[t'
+.
|t'
lt,t,t,
.f,,
r'
lt'
r , r . h e r,.ft ' i s c o n t i n r r o uosn a r r i r r t e r . v a l
.1,,,r,),lr- f,,.1t.,)n.,*,1,.f'(.r)rl.r..
r'orttailrittgtlte trutnltet'srr. 1r.arrrlr'. r'egarrllt'ss
ol tlrt'or'<lt'r.a.|. arrclr,.
5 . I f l ( . r ' ) i s t r n o t l r l f i u r r . t i o nt.h , ' , r / . l ( r ' ) r l . t . - ( l
.l ,,
tj. If ./(.r) is arr even fiul.tion. tlruu
.f
7
I
,f,,'
.l{.,) ,t,,.
I I . l ( . r )] 0 o n l r r 1
. r ]r. h e r rl ' " , , ( , , ,r / . r>, 0
.t,,
8 . I f . q ( . rZ' ). f ( r ) , n l o . b l .r l * , u
22.
",,.1'(.t.)
tlr
Dcfiriti.'
,t.,
7
[ , , " , , { . r ) , 1 , r, [ , , " . 1 { . , . 1
. f D < ' f i r r i t t 'h r t < ' g r : r lt r s t l i . L i ' r i t , f u S r r r r r
S t t l l l r t l s t ' t h a t a f i r t t t r t i o t t . l ' ( . r ' ) i s < o n t i r u r o r r s o r r t h e r . k r s t ' < li r r t e l v a l
l r r . l i ]. D i v i r l e t 1 e
l
:
iritt'rval irrto rr cclrral sulrirrtcrvals.ol length A.r.
" (,h,,,,st,(,ll(, nl1lnl)er.irr caclr
s t t b i t r t t ' l v a l i . t ' . . r ' 1i n t l u ' f i l s t . . r ' 2i r r t l r t , s t , r ' o r r r l. . . . . r ' A . ' i 1t l , , ' A . t h . . . . . ^ r r r l . r . , ,
i, tlr. rrt5.
'
'Ilicu,,lirrr
rl,
r , r ' 1) J r ' -
I
2:1. Funrlarncntal
'flrlollrrr
{ . , ), t . ,
f, , . f
,1 ('ak.uhrs
7b
I
.t
tt.,) ,l.t
,,
l:iltt
1 - ' t rt i. n " l u , r .Ft ,, ( . r ) : , f( . . r ' )
j,
o,+..f',,'
,,,,,,, ri, ,',',r
rtt:,f(q(t.))g,(.r).
f ,"''',,rr,
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 disregarding its direction.
The speeclof a particle irrcrcascs(speedsup) when the velocity and acceleration have
thersarrresigns. The speed clecreascs(slows down) when the velocity and acceleration
have opposite signs.
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). Negative acceleration(deceleration)means that
Tlie acceleration gives the rate at which the velocity is
t[e vgloc:ity is dec:r'easirrg.
crharrging.
Therefore, if .r is the displacernentof a rnoving objec:tand I is time, then:
i)
veloc:itY: u(r) : tr (t\ :
#
i i ) a c'cre l e ra:ti o (t)
n : ."' ( t) : r ' /( /)- #. : #
iii)i'(/)
[n(t1,tt
i v) .r(t)- [ ,,3 1a ,
Notc: T[e av('ragc velclcity of a partir:le over the tirne interval frorn ts to another time f. is
vel;c'itv: T#*frH#:
Average
"(r] -;'itol. wheres(t) is the p.sitionof
the partic:leat tinre t.
25.
The avetage value of /(r)
26
Arca BctwtxrriCtrrvt,s
on [a. ir] is
+,,,,
1,,'
f (r) d:r.
If ./ ancl g are continuousfuncrtionssuch that /(:r) 2 s@) on [a,b], then the area between
,.b
I
I l r e c r r r v e si s / l / ( " ,I - q ( r l ) d r .
Ja
Formulas and Theorems
225
2 7 . Volume of Soiids of R.evolution
Let / be nonnegative and continuous on [a,.b]. and let R be the region bounded above
b y g : / ( r " ) . b e l o w b y t h e r - a x i s , a n d o n t h e s i d e sb y t h e l i n e s r : : n a n d r : b .
When this region .R is revolved about tire .r'-axis.it gerreratesa solid (having circular
fo
crrosssec'tions)u'hosevolume V -
|
{j'(.,'l)2 ,1.,.
/tt
28
Volunrcsof Soli<lswith Knowrr Cross Scctions
l.
Fol cross sectionsof area A(:r:). taken pt'r'lierrcli<'ulartcl the r-zrxis.
',llttttt' :
z.
^rr,
.[rr"
dr.
Fbr <'rossse<rtions
of area A(37)taken perpt'rrriicrrlar
to the 37-axis,
v.ltrttc' -
.[,"
^r,,
rh.
29. SolvirrgDifferential Equations: Graphically ancl Nurnerrir.all.l'
Skrpc Fieicls
Af ever'1'poirrt (.r.r7) a differetrtial ecluatiorrof 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
slope is eclualto the value of the clerivativerat that poirrt. I'hese scgnrerrtsare taugcnt
"vhose
to the sohrtion's graph at the poirrt.
The slope fielcl allows you to sketc:hthe graph of ther solution cul've even though you rlo rrot
have its ec|ration. This is clc-rne
by starting at arry point (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.
Somc t'trlc'ulatorshavtt built in operations fbr drawing slope fields; fcir calculatorsrvithorrt tiris
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 asked to solve ott the AB Calculus Exam can be solved by separating the variables.
R,ewrite the 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 orte c'onstantof inteqration. Use the initial condition to evahrate
this constant.