of(x)=0 o Net-signed Area - WordPress.com
13
Pe+lniteIn.[g[ --- C f(x)7o @ fbf-srq,,,A Y Ateo nSs AX = L=$ n b\ Jr,,* R- fiBrrqyrr"t iuu ft*a^= hZtr*)ax B (:!.
-
Upload
khangminh22 -
Category
Documents
-
view
0 -
download
0
Transcript of of(x)=0 o Net-signed Area - WordPress.com
Pe+lniteIn.[g[
---
C f(x)7o
@ fbf-srq,,,A
Y
Ateo
nSs
AX = L=$n
b\
Jr,,*R-
fiBrrqyrr"t iuu
ft*a^= hZtr*)axB (:!.
2a18-l CALCULUS 2o6111Sec 002 _ ,:rttjutloctober 2018
Ot{,tn 1a410or [.Vs4
5.i1,2 Pl.o1>erties of the Definite lltegr';rl
Page 1
Theorern 5.3
(a) If rr, is irr the rlomain of f , rve definc
Pr, ,,,,, : ,,
@
i ii t li f i. ilrl{'qt'itiill i*t 'tr. it' . tli('ll \i-r' r1,'tiltt
Example 5.4
(;r )
ilr j
8.t'O-.,'2),/',': C,l,IJ'tfl - i),t., : - !rlT34*
0
Theorem S.4 If I nnti .q illc integral:ie on [o. b] and if r is a corrstant. then r:y'.
.f + g, arrd .fg ale integr*ble ou ftl. fi] anri
(d .{j] r:.f (r,)d.r: : ,, f: J(;r,)rJ.r;
rr,i /,jtl,,') *.q( ,J1.," : .{,1,' f t,"),!, - ,!li yt,, ),1 ,'
1r,; .[ft'(.r,) - .,/(r,$r, : .f J (.r')rtr: - .#'.q{r,),/r,
Er;ilrill' l"ii.-,-.i1 I - '', w=t
= -lrn
0.r 0 {
fsa* - sfJil'ax00s -rffi*) --5'q
{x:
Jrx--I+
2018-1 CALCULUS 2061"13. Sec 002 Thursday 11 October 2018
g"lxl
Page 2
Theorem 5.5 If J is integra"bie on a r:lusecl interviri contirinirrg the thrr:e 1:oints o. b.
;trttl r" tlicrt
{o rr.r.)r/.r. : {' y1r.7a.,. , {o s1.,.1n.,.1,, ,1, J,.
&
k/)
a-cb 'l7tO)
.x<O
E4
bcb(rox -- Itcr,'r"
+ lRx)dtx;&C
I^r= t:L
b fl*ta"-/{ -l
: . Ir,ct Jrt, = ICN) dD(
{-l
.t
+Jx axo
tY "Y."1
\",2t
'//lrI
'l /,/tt
t
2018-1 CALCULUS 20671.1. Sec 002 Thursday 11 October 2018 Page 3
rrr!_tr i.ir,{ rl-{.lli },ti
in) ll.l ir iullgt';rlrit, ulr iru" 1,r ;r1t,l ,f t.," i > {} ii,r' ;r.ll l iir ,r" L,. thll
l'l '
I i r., r,l, )- i)I.t tI
ilii lf / ;trr,1 q ;tll itrllgt;tlik, ,,.1r iru.1, ;urri .f t., ) > ,(.r" j ii,r rr]i .r' ir; ,,i. l,l. tlt,.'rr
r', t"I ft.r),1.r') I ,1tti,l,'
.l ,, .l ,,
fi(l.l Tqu) Y'xeta't3
ltV -t
sta
"I[* I =
t'(+ ) =
-[tvl = xL
I{'rxr > bk
{(x) = Zx
Q . 6"t"fi',^Uetls'L ... llortsrlri tXl <-2Y.
F(r) ,6L2_
F(x) = { +L
ftx) = Cos x
CI : 6*fiani*ofrd1 . rri, rrrld ftXl = (ot
F(x) = sin
-Snz. $in
X ..o,^s#
xtCx *4-'x+ 6
2018-1 CALCULUS 706711. Sec 002 Thursday 11 October 2018
5.3.i] The Autiderivative h{et}rorl for Finrling Areas
Page 4
_ L, I I , L.l *,/.).,,,..,,,,, ,,,,r.,1,,,.i,..,ri,-,-,.{ r.r,r- N'F.rt,'rirLtr1,i.. i., 1.-ir''-1.i.,'-il :.r'il*1.f ile;r11 rrnlirlelilirlir,esof J(r')=
tlte gt'iiph of .f olili tlie.r iutelr,al io..i'i" x-lrt'r'e.r'is nnr, point irr tlrt,irrtclral ia.1:l (Fig-urc i.1.lJ. theri
.t/i ir:/', 1: ,-..)
The folloiring exarnple confirurs Fornrula (5.2) irr solne cases x.here a fomrula for A(;r) can be feunrl
using elementary geometr3-.
l(x) &n),ni-'ra x'qr' h}nlftrl $o a 01 X.
I
tljp}/lJ ro a a'r ]
dnri,;&,$"$o R ( bl
il",i'. ;rtjr i fiu,i rii,,,iltir.':rTlr-r , .l'r.,'r ui rlri" rLtlir iuiii.lj,;it.
Solrrtiol
1;r ) .l(.i.) : il; ri : ll Fnlrft&s
fi(x) =g)!mn.,6e
A(x) = 1N
A'G) = 3 -y;)
,1 9*figl\_/
4\,-il.t:/
2018-1 CALCULUS 206111 Sec 002
Liri .ft.i'l: I -- t.: it : -')
Thursday L1 October 2018
Srrl de
A(x1 =L$,y)-- ! (x+z)[x+z)
2_
,v:2*x
4
A't*1 --{h/6<$fup"nl*t
Ac*; do*trnf,"u{"n$ ft'l snx'-2 i', ffi6flsrnnm r{,{t1-ffrI-l-p, finx-- 2 ot X = lD 1tt A0o)
1'.r' t]r, irtiIi,L''tir';rrirr' liL'rltlrl t,r it tli tlt,, riill liti,i,,i ti1r, ,-i'irjr1, ,,i ,i - ,l ,rli,t iiir
irrtelr.al [-1, i].
soruticn (Af*l &r)'niqdtXo.n X= I d., alc)
4'rx) = X-
Otnnlh {uil n'',0 -l d I L
n(-t)$6sui
o
aYn\tilr O .
= I 6-113 +C -.)o'l.i - rrir ^
{
L
a u A(x) "{o>r:rdo'I'6{\t}r}.r ";.Ld;rt1qir1i' X'
A(r) =lt' +C
brltfi \.qr.$o
= L xe o-LZc
'1 d) -L,0flx1 =xz
1ro X---1 ci: x--1 dotpr A(t)
Page 5
(r) = {C,r13.t
l=2-31
II(
I
I
qt"i'\'n
2018-1 CALCULUS 2461.1"1Sec 002 Thursday 11 October 2018 Page 6
dirtr ..i ldirdorvd / 6*finto'bli'',iorvu''5.+ The hrdefirrite Irrtegral
;i.-1.1 The Irrrlefinite lntegralmrqfidi,Jul /rnql{ni nJr '
Tltr'litr,r','ss lf irttrlirig;rrrtirili'ir';rtilcr is L'rri1*r,l anticlifferetrtiation ot'iute6^r'ation. Tl.rt-.. if
, J-t;l--f1;r l.r..i
1i.it
l i., t- ['. T,,r ,'itriiit;r:izC lLi: irti)('r,s]. E{lrtirtil]ti ( ,.;} is 1(,i.ir:I ttsitr.t iltlr,irt';r1 1t,rl;rllr.,1t,
{@-.)r*: r(r) +9
[' retso*rlsryi [^;r ( inkqra't )
b1
tr
ft*t ' arqo{l*ut 0'rte1'-nd)
dry ' arah:r$ dt'h't (d,f{t'''uo'{)
qer'\ tJ3rt*tJ{rnor,vcr vol ftx) = 6x*z
F(^ ) --?*'*?r tEF'0*) '- 6k +L
?*+z'/ +C *f rx*a g6 =
r.,,. ,:.,, il Fl,Ij.;rlt.\ illtli,i|l]'. lilii,'r,i .f..ttt)lliri.rr,1rr,l..l lrtIr,l'\itl. tli1,L 1,'t'
:rtrl irrti5iiittl {'f]t'' i1l.i..i('1j,rl. lr.tt-t 'j-;ll',);rtl;itlfjlir,;'i'.;illlr,ltr tllti jrrlrlr,rl.
I i ; , - i' l,\' r']r,,1']11. 1-lt, ,',,1t-i,,ttl {.' ,rl;1ri,rirt':irlr.lr-.
6.*fiol nUiln
I nrrarnnuro'Pir C )J,
shrf 6 rT r"lv o
fco is-Pr
s&' ,,
--rf' t
114
5.4.2 Integration Formulas
Differentiat iorr Forrrrula Integration Fitrrnula,
1.
'2.
.).
4.
5.
{)-
7.
8.
o
10
1l
72
IJ
t4
t5
1t,
at
18
19
20
d, (('):oI !,:1
-1t), -, l6".ui : ^'or-
,.1
, it'f lr)l: kl'(r)Q:t'
rl
fi1.{t r).r g(.rJ - ['(r\ = g'\.,'1
!tr'l - ,,r'"-l0,.1:
d,,, 1 1, ltlrt.ri : -(l,x- -
.t'
d
- 16r'] : e'rl1 .t
d;iot : rz'l1r o. o > () arrd o,1 I
I .ir-rrl : cosr(1.I'
1i".,*,rl : -si,rdr'd, r .,
d-r ltanJ', : sec- .tr
,l;: lc.rt .t i : -cosec2.r'o,r"
1 ,",'rl - secr tau rrl,.t"
d
J, CO:e 'C,/'_ : -{'{)Se( J'C(tl .t'
1i,,,'"rir... - !cl.r' \/l_ r2d_ 1
-,afL'titu"f :
-
,l t'- "-' |-rL'2rl
;lhr I scc rl: - tatt "i'
d
-,-lrr sin.r'j :cotr
d,., -lli scc.r r tilll.r'j : -cc.r'
il..t'
d
; ilrr lcos"c-r r col J ] : *"t,se".r
1. { odr: C
2. Jkdx:kx*C-tt. fiir.f(*))dr : k I f?)a*
a {lf{") + s(r)ldr : { f(x)dr + J s{r)d,r
r,n+L5.)l r"dr +C,n/-l4r n-f 7
Ct&hlrl +c
7. Ie"dr:ex*C^ (L'
B. Ja'dr -*, *C, a>Aanda,/l
9. / cos rdr : sinr * C
lo. Jsir@@;:-cosr*C
11. / sec2 rdr : tar;.x * C
12. J cosec2xd,r : - cot, u * C
13. / sec rtanrdr : secr * C
14. J cosec r cot rdr: *cosec r -f C
-115. I r:-,-- ndr : arcsin r + C
\/L-r"16. f -! ^ar-arctan.rr(',l*.rz
17. { tanrdr :ln] secrl + C
18. J'cot xdr :ln I sinrl + C
19. J sec rdr :ln I sec r + tanrl f C
20. f cosec rd* : * ln lcosec r + cot r! + C
Ac:rclcuic li-.ar 2018 206111: Ca,1culus 1
2018-1 CALCULUS 206111 Sec 002
Exalnple 5.8
Thursday 11 October 2018 Page 7
(a) J(r:6 - ?.r: * 4)rr,r: : I *'d.x - f+ro\x +
[+dX
= 5{dx -+l*N + [4dx
=[$*.,) -*(+*q) + (rx+ca)
-- {- -C+\x+L={ _t+ qx
?L, -' , '_l , i
1
ilr, L.rl,-'' .tt
@15*= l,xqd,x
=x .+c
ro#lk-4,-hd*oe
I- +2,lnl*l*+,1 +C2- 3 x'
t.) J(/T + 3[),/.,.: If*{ + f{ )d*J
X,I
x'I
x-z
fi--{;=r.u,r -l +t
x?*t , x3
l+L -*+1'Lt t,z .4x t x' tcj2-
27?7t2.X n?lrl 7C
I
{-
T
t{ d* = ,(n lxl +
2018-1 CALCULUS 2A6111. Sec 002
,]'| t - '!.--
Thursday lL October 2018
fdax =
f^* ax =
Page 8
gxt'i +C Ke+c.9n2
1(
aIr' rt
tc.
t,') .l'(Jsiu.,. -r- 2r.r:s.r')r1.i, : - {|COS X + ZSia X + C # flsinr*x=-cotx+c
!or*dx = sin x +C '
'),}(ir lt:' r/i -- 'l
')- \./..
-liJ,i--
L-"t
3Arcsin x - ?ifidwx +cuv-
* t#o\t=?rcsin x +c
I#" =orc*" x4c
2018-1 CALCULUS 2061.73. Sec 002 Thursday 11 October 2018 Page 9
5.5 Integratiou b1 Substitution
Theorem 5.8 Let u be il firirr:tion of ;ti iln{l y' be ;r fiulrtiou r:f ir. Then
ttI l.f(,,)l11rr : f [.f i,r1.,,;;rrl(.r')]r/.r'..t .t
Exarnple 5.9 Evahra*- f$:!loo{O
0 s6,,\fi !kx*,\.;r'"dx ,+r'y'
na,fit,i d = 2x+tJ u'"du
dq =2dx
f^na x
(*n dif&,e,lt,"{
Ic.** r1{o'dx =l [(zx+ ni'o' zd,xL-.nt ffi=ltd"cl
^=(+'61 +c
= -L(z**,1]"1c2 lo1
kt*+r i"Ar \
'=[u'65
$'ilqm\t
T
Sn,\\$ du @ fczr*,J'odx = *-fn*+r1roo2.4x ' lo'
zffi= J [t.x +, )''o d (,zx+ r)Z ) v\^ r-- .*s
+c#(zx+llldu=
