CHAPTER 7 Applications of Integration
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1 CHAPTER 7 Applitions of Integtion Setion 7. Ae of Region Between Two Cuves Setion 7. Volume: The Disk Method Setion 7. Volume: The Shell Method Setion 7. A Length nd Sufes of Revolution Setion 7. Wok Setion 7. Moments, Centes of Mss, nd Centoids Setion 7.7 Fluid Pessue nd Fluid Foe Review Eeises Poblem Solving
2 CHAPTER 7 Applitions of Integtion Setion 7. Ae of Region Between Two Cuves. A d d. A d d. 7. A d d o d d 9. d. se d π π π π. () Intesetion points: A A, nd, d d d (, ) (, ). f g A Mthes (d) (, ) (, )
3 Chpte 7 Applitions of Integtion 7. A d d (, ) (, ) (, ) (, ) 9. The points of intesetion e given b:. The points of intesetion e given b: when, A g f d d (, ) (, ) when, A g f d d d 9 (, 9) (, ). The points of intesetion e given b: nd nd A d Note tht if we integte with espet to, we need two integls. Also, note tht the egion is tingle. (, ) (, ) (, ). The points of intesetion e given b: 7. The points of intesetion e given b: when, A f g d d d 9 (, ) when, A g f d d 9 (, ) (, ) (, )
4 9. A f g d. d (, ) (, ) (, ) (, ) Setion 7. Ae of Region Between Two Cuves A d ln ln ln ln.9 (, ) (, ) (, ) (, ). () The points of intesetion e given b: (, 9) (, ) (, ) () Numeil ppoimtion:.7.7. when,, f g d g f d A d d d 7 d. () 9 The points of intesetion e given b: (, ) (, ) () Numeil ppoimtion:. when, A d d 7. () f, g The points of intesetion e given b: (, ) (, ) () Numeil ppoimtion:.7.9. (, ) (, ) B smmet: A when ±, ± d d d d
5 ( Chpte 7 Applitions of Integtion 9. (). () (, (, ( (, ) (, ) The points of intesetion e given b: ± A f g d d (, ) nd () on, You must use numeil integtion beuse does not hve n element ntideivtive. A d.79. tn () Numeil ppoimtion:.7 A ln..7 f g d. sin tn d os ln os π π, (, ) π π, g f A os os d os d sin (, ) g ( π, ). π π f π 7. A e d e e. ( ), e (, )
6 Setion 7. Ae of Region Between Two Cuves 7 9. () A sin sin d () Numeil ppoimtion:. os os. () (, e) A e d () Numeil ppoimtion:. (,.) e e e. () The integl A d () A.77 does not hve n element ntideivtive.. () The intesetion points e diffiult to detemine b hnd. () Ae os d. whee.. 7. F () F t dt t t F () F t t t 9. F os d sin sin () F F. () F. θ θ θ
7 Chpte 7 Applitions of Integtion. A 9 d d 7 7 d 7 d = 9 (, ) = (, ) (, ) =. Left bound line: Right bound line: A d d (, ) (, ) (, ) (, ). Answes will v. If ou let nd n, b. () Ae 9 sq ft Ae sq ft 7. f f At,, f. Tngent line: o The tngent line intesets f t. A d 7 f () = (, ) = (, ) 9. f f At,, f. (, ) = + f () = + ( ), Tngent line: o The tngent line intesets f t. A d tn.
8 Setion 7. Ae of Region Between Two Cuves 9 7. on, A d d You n use single integl beuse (, ) (, ) on,. (, ) 7. Offe is bette beuse the umulted sl (e unde the uve) is lge b 9b A 9 d 77. Ae of tingle is 9 b d 9b 9 b d 9 lim 9 b 9b 9 9 b 9 9 b 7 9 b 9 b 9 9. whee nd i i is the sme s n n. n i i i d. ( 9 b, b) (... 9 b, b). f () (, ) Sine < <, selet.7. B O (, ).... OAB d ± A.t 7..t 7..t dt.t dt $. billion.. () 7.. t 7.e.9t R t 9.97e.7t E () Reeipts (in billions) Time (in es) t 7 Suplus dt 9. billion dolls (Answes will v.) Ependitues (in billions) Time (in es) (d) No, > foeve beuse. >.. No, these models e not ute fo the futue. Aoding to news, E > R eventull. t
9 ( Chpte 7 Applitions of Integtion. %: P 9,e.t %: P 9,e.t Diffeene in pofits ove es: Note: Using gphing utilit, ou obtin $9,. e 9,e.t 9,e.t dt 9,.t e.t.. 9,...7 9,. $9, 7. The uves inteset t the point whee the slope of equls tht of,.. k... () The vlue of k is given b... k k... Ae d... d () A..9 V A.9. m () V. 9, pounds 9. Tue 9. Line: A sin 7 d os We wnt to find suh tht: b d b b b b But, b b beuse b, is on the gph. b b b b b b b 9b (, ) π ( 7 π, π = b 9 (b, )
10 Setion 7. Volume: The Disk Method Setion 7.. V. V. V Volume: The Disk Method d d d d d d V d d V d d., (), R, V d d R, V d () R, V d d (d) R, V d d 9
11 Chpte 7 Applitions of Integtion., inteset t, nd,. () R, V d d R, V d d. R, 7. V d d R, V d d ln ln ln. 9. R,. V d d R, V d d
12 Setion 7. Volume: The Disk Method. R,. V d ln ln. d R, 7. V d R e, V e d e d e e. 9.. The uves inteset t, nd,. V d d 77 V d 9 9 d 9 h, Volume of one d d. V sin d sin os d (, ) Numeil ppoimtion:.9
13 Chpte 7 Applitions of Integtion. V e e e d 7. V e d Numeil ppoimtion:.9 e d.9 9. V tn. d.. sin d epesents the volume of the solid. geneted b evolving the egion bounded b sin,,, bout the -is. A Mthes () π π. The volumes e the sme beuse the solid hs been tnslted hoizontll. 7. R, 9. V d Note: V h R, V d d = (, ) (, )
14 Setion 7. Volume: The Disk Method. R H H, H, h V H h d H H d H H h H h h h h H H h h H h H. V d d. () R, V 9 d d R, 9, V 9 9 d Totl volume: V Volume of wte in the tnk:, ft d d, When the tnk is one-fouth of its pit:,,, 7, 7, 7. Depth: 7.. feet When the tnk is thee-fouths of its pit the depth is. 7. feet.
15 Chpte 7 Applitions of Integtion h b 9. () d (ii) (iv) b d () d b (iii) is the volume of ight iul linde with dius nd height h. is the volume of n ellipsoid with es nd b. is the volume of sphee with dius. = ( h, ) (, ) = b = (d) h ( b, ) ( b, ) (, ) (, ) (i) h d (e) R R d (v) is the volume of ight iul one with the dius of the bse s nd height h. = h ( h, ) is the volume of tous with the dius of its iul oss setion s nd the distne fom the is of the tous to the ente of its oss setion s R. R + R R. Bse of oss setion () A b V d A bh V d
16 Setion 7. Volume: The Disk Method 7. Bse of oss setion () () A b V d A bh V d d (d) A V d A b V d b. V R R R d 7. V R R d d R R R R R R R R
17 Chpte 7 Applitions of Integtion 9. V d 7. V d 7. V d 7. () When epesents sque. When : : = = epesents ile. 77. () R R ± V R R d R d R d A d d To ppoimte the volume of the solid, fom n slies, eh of whose e is ppoimted b the integl bove. Then sum the volumes of these n slies. R d is one-qute of the e of ile of dius,. V R R Setion 7. Volume: The Shell Method. p, h. V d p, h V d d. p, h 7. V d p, h V d d
18 Setion 7. Volume: The Shell Method 9 9. p. h V d p, V e.9 h e d e d e e. p, h V d d. p nd h if <. p V nd h if. d d 7. p, h 9. V d d p, h V d d (, ) p, h V d d
19 Chpte 7 Applitions of Integtion. p, V d 9 d h. The shell method would be esie: d shells V Using the disk method: d Note: V V 7. () Disk R, V d Shell p, V d h () Shell p, V d d h 9 9. () Shell p, h V d d (, ) (, ) Sme s pt () b smmet CONTINUED
20 Setion 7. Volume: The Shell Method 9. CONTINUED () Shell p, h V d d (, ) (, ). Answes will v. () The etngles would be vetil. The etngles would be hoizontl.. This integl epesents the volume of the solid geneted b evolving the egion bounded b,, nd bout the -is b using the disk method. d epesents this sme volume b using the shell method. d d Disk method. (). 7. () 7 / / = ( ) = ( ) ( )...,, V d. V d e,,, Volume 7. Mthes (d)
21 Chpte 7 Applitions of Integtion. p, Now find h V suh tht: d d d totl volume. ± (Qudti Fomul) Tke.7, sine the othe oot is too lge. Dimete:. V d d d d. () d sin os C os sin os sin d Hene, sin d sin os C. (i) p, h sin V sin d sin os (ii) p, h sin sin sin V sin d sin d sin os π π π π.. π π π π π.
22 7. d () Plne egion bounded b,,, Revolved bout the -is d 9. Setion 7. Volume: The Shell Method d () Plne egion bounded b,, Revolved ound line = Othe nswes possible Othe nswes possible. Disk Method R V d h h h h. () b Ae egion b n n d R n bn nn b n b lim R n lim n n lim n bn b b n n n b b n bn n b n n n n bn n n n n () Disk Method: (d) b V b n n d b R n lim R n lim n b n b n b n n d n n b bn b n nn b b n n n b n n n n n n n lim b b n n (e) As n, the gph ppohes the line.. () V f d,7 ubi feet CONTINUED
23 Chpte 7 Applitions of Integtion. CONTINUED Top line: Bottom line: V,,,7 ubi feet d d d d (Note tht Simpson s Rule is et fo this poblem.) 7. 9., () V d d V d V V d 7 7 V V d d 9 () V d d ields no volume.
24 Setion 7. A Length nd Sufes of Revolution Setion 7. A Length nd Sufes of Revolution.,,,.. () d s d, s d.9, s d d d , b s d d d 779. lnsin,, os ot sin ot s s s d ln s ot ln ln.7. e e. e e,, e e,, s e e d e e d e e e e.7, d d s d d d 7
25 Chpte 7 Applitions of Integtion. (), () L.7 L d 7. (), () L.7 L d 9. () sin, os () L. os L os d.. () e, () L. ln e. L e d Altentivel, ou n do ll the omputtions with espet to. () e, d d e () L. d d e L e d. () tn,.. () L.7 L d
26 Setion 7. A Length nd Sufes of Revolution 7. d d s Mthes d (, ) = + (, ) 7., () d. d 7 7 () s d 9 d. (Simpson s Rule, n ) (d).7,. 9. () f No, f is not defined. () f f Divide, into two intevls. s 9, : d 9 9 d, s s 7 d 7., : s 9 d 9 9 d,
27 Chpte 7 Applitions of Integtion. () (),,,,,, L d.7, L L L 9 d.79 d.9 d.. When,. Thus, the fleeing objet hs tveled units when it is ught. s d The pusue hs tveled twie the distne tht the fleeing objet hs tveled when it is ught. d. osh, 7. sinh sinh osh L osh d sinh sinh 7. m osh d s 9 9 d sin sin sin sin.9 9 d 9..,, S d d 9. 9,, S d d 7 7
28 Setion 7. A Length nd Sufes of Revolution 9..,, S d 9 9 d 9 d sin os,, S. sin os d 7. A etifible uve is one tht hs finite length. 9. The pelulus fomul is the sufe e fomul fo the ltel sufe of the fustum of ight iul one. The epesenttive element is f d i i i f d i i i i.. h h. 9 9 h S h d h h 9 S 9 d 9 d 9. See figue in Eeise S 9 d 9 d Amount of glss needed: V 7.. ft. in. 9 d 7 ft. ft. in.
29 Chpte 7 Applitions of Integtion 7. () We ppoimte the volume b summing si disks of thikness nd iumfeene equl to the vege of the given iumfeenes: The ltel sufe e of fustum of ight iul one is sr. Fo the fist fustum: () V i S.. 9 Adding the si fustums togethe: S i i ubi inhes C i C i i (d) S C i V d 7.9 ubi inhes d 79. sque inhes h R s 9. () V b d S = b b b b d d d b b() lim V lim b b b (d) Sine > we hve b b d > nd lim ln b. Thus, b b lim d b. > on, b d ln b ln b
30 Setion 7. Wok. Individul pojet.., S d d 9 [Sufe e of potion bove the -is] k, k k h kw k h w h w w h B smmet, C w d. w h (w, h) w 7. Let be the point on the gph of, whee the tngent line mkes n ngle of with the -is. (, ) = 9 9 L 9 d 7 (, ) Setion 7. Wok. W Fd. ft lb W Fd joules (newton-metes). Wok equls foe times distne, W FD. 7. Sine the wok equls the e unde the foe funtion, ou hve < d < < b. 9. F k. F k k k W 7 d 7 in. lb. in. lb. ft lb k k W F d d 7 n m 7. joules o Nm
31 Chpte 7 Applitions of Integtion. F k. k9 k 9 W 9 d 9 in. lb ft lb W k d k k 7 W d 7 Note: inhes foot k 7. ft lbs 7. Assume tht Eth hs dius of miles. F k k k,, F,, () W 7. mi tons. 9 ft lb W,, d,,,, d 9. mi ton.7 ft ton 9. Assume tht Eth hs dius of miles. F k k k,, F,, () W W,,,, d,,,,.7, 9,. mi ton.9 mi ton. ft lb,, d,,,,.,,. mi ton. mi ton.7 ft lb. Weight of eh le: Distne: () W. d 99 9 ft lb W.. d ft lb. Volume of disk:. Volume of disk: Weight of disk of wte: Distne the disk of wte is moved: 9 d 9, d W 9, 9 Weight of disk:. Distne: W. 9 d 7 9, 7, newton metes ft lb
32 Setion 7. Wok 7. Volume of disk: Weight of disk: Distne: W. d. d..,7. ft lb 9. Volume of le: V lwh 9 Weight of le: Distne: W 9. W. 9 d.. 9. d 9. d Tto The seond integl is zeo sine the integnd is odd nd the limits of integtion e smmeti to the oigin. The fist integl epesents the e of semiile of dius Thus, the wok is. W 7 ft lb.. Weight of setion of hin: Distne: W d 7. ft lb. The lowe feet of hin e ised feet with onstnt foe. W ft lb The top feet of hin e ised with vible foe. Weight pe setion: Distne: W d ft lb W W W ft lb. Weight of setion of hin: 7. Wok to pull up the bll: W 7 ft lb Distne: Wok to wind up the top feet of ble: foe is vible 7. W d 7. Weight pe setion: Distne:.7 ft lb W d. ft lb Wok to lift the lowe feet of ble with onstnt foe: W 7 ft lb W W W W ft lb
33 Chpte 7 Applitions of Integtion 9. p k V k k W V dv ln V. F W k k k d k k units of wok ln.9 ft lb. W. ln d 9. ft lb. W d,. ft lb Setion 7. Moments, Centes of Mss, nd Centoids () L feet , 9, 9 m (, ) m (, ),, m (, ) m m m (7, ) (, ) (, ) m (, ) m (, )
34 Setion 7. Moments, Centes of Mss, nd Centoids. m M M m M d d d (, ) M m,,. m M M M m M m d d d d d 7 7 (, ) (, ),, 7. m M M M m d 9 d 99 d 9 99 d d d 7 (, ) (, ) M m 7 9,,
35 Chpte 7 Applitions of Integtion 9. m M d d M m M d 9 (, ) M m 9 9,, 7. m d M M m d (, ) B smmet, M nd..,, m M M m 7 9 M d d d d 9 7 d d 7 (,) (, ) M m 7 9,,. A d M M d d 7. A d 9 M d d 7 M d
36 Setion 7. Moments, Centes of Mss, nd Centoids 7 9. m M d,,7, M d d,. 9 d. d M m. M m. Theefoe, the entoid is.,... m d 9.7 M d d.7 M m. b smmet. Theefoe, the entoid is,... A A b b b b, b, b b b b d b d b d d d = b + ( + ) (, ) Fom element geomet, b, is the point of intesetion of the medins. (, ) ( b, ) = b ( ) (, )
37 Chpte 7 Applitions of Integtion. A b A b b Thus, b b d b b b b b b b b b d b b b b b b b b b b b b b b b b b, b b, b b b. The one line psses though, nd, b. It s eqution is b The othe line. b psses though, b nd, b. It s eqution is b., is the point of intesetion of these two lines. b b b d b b b b d (, ) (, ) b = b + (, ) (, ) (, b) 7. b smmet. A b 9. () A b b b b,, b b smmet. b M b d b b () b d beuse b is odd. b = b (d) > b sine thee is moe e bove b thn below. (e) b b M b b b b b b b b b d b d b b b b b b A b d b b b b bb bb bb M A b b bb b
38 Setion 7. Moments, Centes of Mss, nd Centoids 9. () b smmet. A () M,, M A M A,97..,,. f d f d 7, 9 7, (Use nine dt points.) 7 7,. Centoids of the given egions:, nd,. Centoids of the given egions:,,, nd Ae: A,,., Ae: A 7,,,.97 7, 7, 7 7. Centoids of the given egions:, nd, 9. is distne between ente of ile nd -is. A Mss: is e of ile. Hene, V A 79..,,.,. A d (, ) V A.
39 Chpte 7 Applitions of Integtion. m m... m n M m... m n n M m... m n n M m, M m. () Yes. Yes. () Yes.,, (d) No,,,,, 7, 7. The sufe e of the sphee is S. The length of C is s. The distne tveled b the entoid is d S s This distne is lso the iumfeene of the ile of dius. d. Thus, nd we hve. Theefoe, the entoid of the semiile is,. (, ) 9. A n d n n n M M m A n n d n n n d n n n n = n (, ) M m n n M m n n n n Centoid: n n, n n As n,,,. The gph ppohes the -is nd the line s n. Setion 7.7 Fluid Pessue nd Fluid Foe. F PA. 9 lb. F.h.h. 7. lb. h 7. L F. d 9. d 9.. lb h L F. d lb d
40 Setion 7.7 Fluid Pessue nd Fluid Foe 9. h. L F. d. d..9 lb h L F 9 d 9 7, newtons. h. h L 9 F 9 d ,, newtons 9 L F.7 d 7 d 7 lb 9 7. h 9. h L F.7 d. d. 7. lb L F 9 d 9 d lb. h k L F w k d w k d d wte level The seond integl is zeo sine its integnd is odd nd the limits of integtion e smmeti to the oigin. The fist integl is the e of semiile with dius. F w k wk
41 Chpte 7 Applitions of Integtion. h k L b h F w k b d h wb k h h wbhk wkhb k wte level h b b h. Fom Eeise : F 9 lb 7. h F. L d Using Simpson s Rule with n we hve: F lb 9. h L F. d.7 lb. () If the fluid foe is one-hlf of. lb, nd the The pessue ineses with inesing depth. height of the wte is b, then h b L b F. b d. b b d. d F Fw w b b. b b. b. b. ft.. hl d, see pge.
42 ) Review Eeises fo Chpte 7 Review Eeises fo Chpte 7. A d. A d (, ) (, ), (, ) tn,, (, ) (, ). A d 7. (, ) (, ) (, ) A e e d e e e (, ) (, e ) (, e ) 9. A sin os d os sin. A d d 7.7 (, ) (, ) ) π, ) π π, ).. A d d (, ).7 ± A d A d d d (, ) (, ) (, )
43 Chpte 7 Applitions of Integtion 7. A d d d d (, ) (, ), A d d (, ) 9. Job is bette. The sl fo Job is gete thn the sl fo Job fo ll the es eept the fist nd th es.. () Disk V d Shell V d () Shell V d d (d) Shell V d d
44 Review Eeises fo Chpte 7. () Shell V d Disk V d 9. Shell V d d tn 7. Shell: V d u u d u du V d u u du u u u u du u u u du u u u ln u 9 ln.9 9. Sine, A d. u u d du A u u du u u du u u
45 Chpte 7 Applitions of Integtion. Fom Eeise () we hve: Disk: d 9 V ft V 9 9 d B Newton s Method,. nd the depth of the gsoline is..9 ft.. f. f f u u d u du s d uu du u u du u u u u.7 osh, sinh s sinh d 9 sinh d. ft b Simpson s Rule o gphing utilit 7. S d 9. F k k F W d in. lb.7 ft lb (, )
46 Review Eeises fo Chpte 7 7. Volume of disk: Weight of disk:. Distne: 7 W. 9 7 d. 9 7, ft lb. ft ton. Weight of setion of hin: Distne moved: W d ft lb. b W F d d.7 7. A d d A d d d d (, ),, 9. B smmet,. A d A d d (, ),,
47 Chpte 7 Applitions of Integtion. b smmet. Fo the tpezoid: m M Fo the semiile: M m Let u, then u nd d du. When, u. When, u. M u u du u u du u du Thus, we hve: d d d d u The entoid of the blde is , = + (, ) 7 (, ). Let D sufe of liquid; F AeD Aedepth of entoid d d D f g d d f g d D weight pe ubi volume. d D f g d f g d d f g d d f g d D d g f (, )
48 Poblem Solving fo Chpte 7 9 Poblem Solving fo Chpte 7. T R d lim T lim R. () V V R R ± V R R V R d d d Integl epesents e of ile R R d R d. V d h h h h whih does not depend on! + = h h
49 Chpte 7 Applitions of Integtion 7. () Tngent t A:, Tngent t A, : 9. To find point B: Tngent t B: To find point C: Ae of Ae of R C, d 7 S, Ae of S e of R s ft dt () s ds f d ds f d () (d) B, d e S e R ds f d d d d d d s t dt s 9 t dt 7 9 t 7 This is the length of the uve fom to.. () b smmet M m d d 7, 7, 9 t dt d To find point B: Tngent t B: To find point C: Ae of Ae of C, R d 7 S Ae of S e of R. 7 () m M b lim lim b b B, d d b b b d b b b b b b b b b, b b,,,
50 Poblem Solving fo Chpte 7. () W e W e 7. Point of equilibium: P,, Consume suplus. d Podue suplus. d.., p 7. We use Eeise, Setion 7.7, whih gives F wkhb fo etngle plte. Wll t shllow end Fom Eeise : F. 99 lb Wll t deep end Fom Eeise : F. 9,9 lb Side wll Fom Eeise : F. 9,9 lb F. d d = = =, lb Totl foe: F F,9 lb
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