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**Unformatted text preview: **Page 2&6 Chapter 6, Additional Appficatians of the Intagraf CHAPTER 6 Additional Applications of the Integral 6.1 Area Between Two Curran, Pagw 3ﬁ1—362 :1
1. —za+ﬁ-x-—5=fx-i} _- _0ﬁ
29-9:+T=ﬂ 9-"- my?" 1112' - 1.-
jain 2: dr+ j (—sin 2:] d2:
0 m _ _1 “I” _1 I _
_.( Jparish)” 4§m2=L!2—1+1 7f: III—9+ﬁz-ﬁ) — (ga— W: 1 4- (”‘13353-1
7!: (:41}3—(£—1)=|J = iiqugx‘gﬂd‘ {xv—1][I[r— 1}E — 11:0 1.
=(_%g+§!g_§l)‘7iﬁ:% (m—l)[r-—l—-lj[x-1+lj:0
5:0.1, =3 -ﬁ.1, Aral Huh-rem Two Curves P131 24!? . 112-'5§=0 Thecurveay=r=andy=3interaectnt
' (n. a) and (1, 1}.
“F_E]=ﬂ n ‘ - 1
: 9:0.5 A=I{:!-:)d=+£(zw::)dz
ill-1' IE: '1 gun I :I_
1:: I »="
a Jamal 1 I: IrJ'IIIh-IFIIIIIIH—l
z... «mania E..- B ' Jaws! - #2,“!
5:125 a
Fig“? I: T
IF—Ey=-y Themy=§andy¥zintemct at.
[-11 —1)’(ﬂ’ a), Hit]. (1. 1}I A = 21;: — :3) it ﬂy alumna Page 243 Chapter l5. Additional Applications of the Integral =(§. _ 1,111]; 12 The y: :1:2 — I] and y = D cum-u interned 51:61.0].
3.” a
A=J'[EI — 491a: +J(4£ - dez'
u 3;:
= {9: — 45’3”? + (i=3 - 92H;
= 2? 13. l].
The. names inberacct. at fl]. El]
and (— a. 54)and{3,5‘1].
A = 2TH]? - '1] it By inhuman!
Thecurvaay=z2rlandy=ﬂintremectat u
{—1.ﬂ)and [1, a).
=2(§x3 — 1&3]:
1 2
A=J[l—:3]dr+J(E-l]dz =§E£
—1 1 4:3 + 2H; + G13 — a”: i
a
E 7-. (5.1. Area Between Tm Curves Page 249 16.
- m 1 W11
‘ Junmum l1nl!‘.!|}‘;ff}|
1: 'IHIilmﬂiIil' “VIHIII H W
M “I“ il‘i "
'|
i
Hecurvesinlemcctaﬂt'B} £2+3t—E=—:i+z+'f
'“I'ﬂ' 215+2x—12=D
22103 - 3r" - ﬂ r1]: Bum-mm {H 2“”:1 _3
n The curve: intersect. at [2. E]
and {-3, — Ii].
21:? {r213 — 211+ 11M:
—5
= 41.3- 9+ 12:][33
.. 125
" T The curves interment. at. {0, - 1]
[—2, — 11] and [3. 59). Page 250 Chapter 6, Addﬂiona! Appﬁcatim: oft-He Integral U
A=I[23—l:2—6z)dz a
+ I[—:a+zz+6z) dz
U = [37:4 — 3'59 _ 3:?)[32 itiiillﬂhhlﬁ The curves y = sin x and y = m a: intersect “5) 3“ G'T -
1r,“ A: J-(coax— sinx)dt
[I = (sin a: + cos 3: ”TE!“ Thecurves y=ldx —1|a-l1d y: a: - 5 r--
nol. intersect on [I], ll]. but the absqute val- —
function causes the equation of the line to
change at 3:: U4. 134
The curveu y: sin rand y = sin Exititerscct A =I(—4z+1-— :2 + 5) d:
when a: = o, «fa. and iv. ° 4
m 1, +[(4:—1-z‘+5)ds
21 = J (sin 2z—uin z] ds+ [{sill x—sin 2):] d'x ”4
a aria P 25.!
.=. 16.1. Am Batman Two Cum '5' {-2E+ x — §13+521|1121
+{23’ — x — 59+5:}L1L 323
_ Tf w .um‘ﬂlﬁiﬂ: '- (=5— 29- £+ 214:
-—1 2
+I(—9+29+=—2}a
l 1&1"— ga— éjz’ + 2:3le
+ (—‘zz‘+ §r‘+ £4? mil? 1 - cuwminhersect at (£1.21. [0, —2},anc} JI—
(33-3y2-4y+12)dy
-2 - '+T(—y3+:sy"+4y-12)ay 2 Page 252 152+Ex+l=4—4za
§§+2:—3:0
(55- 3)(-r+1)=|]
:=E,—rl [n the domain, the curves intersect at 1:: 0.5 0.6 ”:3 1 ' ﬁr” III 0.6 =[2ln(:+ 1] - sin—1:” = 2 In 1.6 — sin-10.6 If y = f(x)ar1d y = 9(2) am given, it is best
to use vertical strips because one need not
solve for z = f _1(I:)m1d z = y'_ 1(z]. Tiltl’c
are usually fewer integrals involved. For the
same reason. if 2: = If“), x = 3(3), use
horiznn‘al strips. 11'? * We want
I{1_ sin y]dy=% I {1 — sin y) a’y
a 0 {M"msr]|;=%[ar+msvllwé:2
(k+cmk) — 1=§KE+°J — (ﬂ+1)]
k+mek=§+%
I: a: 0.34 By calculator 23. The ”nae intersect at a: = —1;the|ine
2y = 11 — a: intersects the parabola aL I: = 3 and —3.5: the line y = ?r+13 intersects the parabola at .1: : —2 and a: = 9. Chapter 5, Additional Appﬁﬂ Eon: of the In: _ A: [7:4—13—(12 -5)]d= 3
+][g(11—x)-(£—a)]dz
—1 = g3+1a= - §£+54|3 Might-é:2 —§:3+521|_31 =3l+933=léﬁ mass This is anmcigllLl: of a circle with radiun A = :‘rﬂv’ﬁf =
so. Since V’s-wig: 1 implies y =1—2zla" '
‘1 ,1 =J(1— 23“ + 2) dz:
0 =<= — WHFJIH
31. n. A horizontal strip has area d21=2tdy=21f§= - 1.7113} These strips lie between 1; = — I: and y - ...

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