Calculus Third Editon By Strauss, Bradley and Smith sec6.1

Calculus (3rd Edition)

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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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