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# sol3 - MATH 138 Calculus 2 Fall 2010 Solutions#3 ilﬁ A...

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Unformatted text preview: MATH 138 Calculus 2 Fall 2010 Solutions #3 ilﬁ A cmss‘section is a disk with radius 1 - :2. so its area is 14(3): 17(1 — 32?. I I = 2.- 1.: __ 223.- v Lam Lara IJd l =27r] (1—2mz+m*)dr=2nlr—§Ia+ém5]; o it,” A crass-section is a disk with radius 2 ﬂ so its area is A(y)=?r(2\/17) i3: !|A(y)riy=ﬁ\$iﬁ(2\/§j‘ldy=dnﬂnydy ﬂ = 41r[%y2]: 2 2mm = 162::- iU-Ir) A cross-section is a washer with inner radius ﬁr“ and outer radius 5 — :52. so its area is A(a:) = «(5 — m")a — ram“): 2 11125 — 10::2 + 3:" — 1139:“). v = If? A(:) dz = ff, «(25 — map2 + %:") dz = 27rf: (25 — 10:“ + ﬁx‘) dz: = 214251: — \$13 + 3:35:35]: = 21r(50— 5:? + 6) = 1—33? 11d] A cross-section is a washer with inner radius a: z 2 V6 and outer radius 2, so its area is Aty) = «[9? — (2 ﬁ)“] ' = #01 - 4y) = «(1 — y). V = fol Aty) dy = f; M1 — y) dy = 4n — is]; = 4«[(1— a) — o] = 2» 2U) Jeromm =£_mf—— (3:1 ——dz.Now l 1 [(374-1sz 3f—du [u=3:r:+l.du=3dx] ——ﬁ+C——m+c, . 1 1 _ _ 5"" .1153, [—3—(33:+ 1}]:— 111.“; ['3(31+ 1) +5] ‘0+ 12 ' 12‘ comment 2U») f 1 we2:1—5“':‘=._‘.i‘_’”.,‘,t 2I_5d3=,iiﬂfm[imih-5I]?=‘l_iglo[%ln5—ﬁlnl2t—51]=—oo Divergent '2' it) If: :12": do: = film If“: dz + f0” an”? dz. If“); 82—:- dz: lim (__)[e -2310: lim (——)(1-—e_ﬂ) =_%.1 =—§.and 1... 0° 1' i—--— won °° “rada- —r: i 4’ 1 1 In xv ,Lwt —;)[e a airy—1 )(e “1) =—§-(—1)=§. Therefore. ff; 6” six: —-;— +% = . Convergent 2M) l-um £—-uo If: ainﬂdB = {1113“ f: sinﬂdﬂ *— lim [- cos 6]; = lim {— coat + 1). This limit does not exist, so the integral is divergent. Divergent - _ 92¢) \/_._(i;1=1 li1n j: {3-—x .12)_”2d.1 -:= lim [—2L3~£)U21;=—2 li1ln (m— VT)" —‘2(U-l)= '2 . .. 1- 2 3_\/—-.; l- ,3— t—ud Convergent _ _ . _ ———- _ _ _ _ _...—-7" .1. 9H,“) "For 17")O(769+l < 19:31:; 00 . SL'iZd'X M WWV‘M wa‘fk P127 1 ) 50 72th) . 2 + e" 2 . ., l °° 1 . . . . . For: _>_ 1. a: > ; [Since a > 0] > Fr" 2? d: IS divergent by Equation 2 With 11 = l 5 1, so I °° 2 + v" . . ‘ . T M ls divergent by the Comparison Theorem. I ‘ 11" 7651-; 7 21" DC -r/ => 0 4 —9E~n._[.. s q 7!. xL-t- I ...
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sol3 - MATH 138 Calculus 2 Fall 2010 Solutions#3 ilﬁ A...

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