CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS

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CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS 6. VOLUMES USING CROSS-SECTIONS. A() ;, ; (digonl) ˆ Ȉ È V A() d d c d 6 (dimeter) c d c d c ˆ 6. A() ;, ; V A() d d. A() (edge) È Š È Š È ;, ; V A() d d 8 ˆ c 6 (digonl) Š Š c ˆ 8. A() ;, ; V A() d d. () STEP ) A() (side) (side) ˆ sin Š Èsin Š Èsin ˆ sin È sin STEP ), STEP ) V A() d È sin d È cos È( ) È () STEP ) A() (side) Š Èsin Š Èsin sin STEP ), STEP ) V A() d sin d c cos d 8 (dimeter) 6. () STEP ) A() (sec tn ) sec tn sec tn sin sec sec STEP ) cos, Î ˆ sin ˆ Î cos cos Î STEP ) V A() d sec d tn c Î È È Š Š Š Š È ˆ ˆ () STEP ) A() (edge) (sec tn ) ˆ sin sec cos STEP ), Î STEP ) V A() d ˆ sec d Š È È cî sin cos 7. () STEP ) A() length height 6 6 STEP ), STEP ) V A() d 6 d c 6 d 6 6 Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

8 Chpter 6 Applictions of Definite Integrls 6 () STEP ) A() length height 6 Š 6 6 9 STEP ), STEP ) V A() d 6 9 d c d 8 6 8. () STEP ) A() se height ˆ È 6 6È STEP ), STEP ) V A() d ˆ Î 6 d Î 8 È Î 8 Î () STEP ) A() ˆ dimeter Š ˆ STEP ), 8 8 8 STEP ) V A() d ˆ Î d Î ˆ 8 6 6 9. A() (dimeter) Š È ; c, d ; V A() d d d c ˆ Š 8 d ˆ 8 c c. A() (leg)(leg) È ˆ È ˆ È ; c, d ; V A() d d. The slices perpendiculr to the edge leled re tringles, nd similr tringles we hve h Ê h. The eqution of the line through, nd, is, thus the length of the se nd the height ˆ 6.Thus A se height ˆ ˆ 6 6 6 ˆ nd V A d 6 d 6. The slices prllel to the se re squres. The cross section of the prmid is tringle, nd similr tringles we hve d 9 9 h c Ê h. Thus A se ˆ Ê V A d d. () It follows from Cvlieris Principle tht the volume of column is the sme s the volume of right prism with squre se of side length s nd ltitude h. Thus, STEP ) A() (side length) s ; STEP ), h; STEP ) V A() d s d s h h () From Cvlieris Principle we conclude tht the volume of the column is the sme s the volume of the prism descried ove, regrdless of the numer of turns Ê V s h Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

Section 6. Volumes Using Cross-Sections 9. ) The solid nd the cone hve the sme ltitude of. ) The cross sections of the solid re disks of dimeter ˆ. If we plce the verte of the cone t the origin of the coordinte sstem nd mke its is of smmetr coincide with the -is then the cones cross sections will e circulr disks of dimeter ˆ (see ccompning figure). ) The solid nd the cone hve equl ltitudes nd identicl prllel cross sections. From Cvlieris Principle we conclude tht the solid nd the cone hve the sme volume.. R() Ê V [R()] d ˆ d Š d 8 ˆ 9 6. R() Ê V [R()] d ˆ d d 8 6 7. R() tn ˆ ; u Ê du d Ê du d; Ê u, Ê u ; Î Î Î ˆ V [R()] d tn d tn u du sec u du [ u tn u] ˆ 8. R() sin cos ; R() Ê nd re the limits of integrtion; V [R()] d Î Î (sin ) (sin cos ) d d; du d u Ê du d Ê ; Ê u, 8 u ˆ 8 8 8 6 Ê u Ä V sin u du sin u d 9. R() Ê V [R()] d d Î. R() Ê V [R()] d d d ( 8 7 7 Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

Chpter 6 Applictions of Definite Integrls. R() È9 Ê V [R()] d 9 d c c 9 7 9() 8 6. R() Ê V [R()] d d d ˆ (6) Î Î È. R() cos Ê V [R()] d cos d csin d Î ( ). R() sec Ê V [R()] d sec d Î Î Î cî cî c tn d Î [ ( )] Î. R() È sec tn Ê V [R()] d Î Š È sec tn d Î Š sec tn sec tn d È Î Î Î Œ d È sec tn d (tn ) sec d Î Œ[] È Î tn [sec ] Î ˆ È È Š Š È Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

Î 6. R() sin ( sin ) Ê V [R()] d Î Î Î ( sin ) d sin sin d ( cos ) sin d ˆ cos sin Î sin cos Î ˆ () (8) È c c 7. R() Ê V [R()] d d c d [ ( )] Section 6. Volumes Using Cross-Sections Î 8. R() Ê V [R()] d d 9. R() È sin Ê V [R()] d Î sin d ccos d Î Î [ ( )] c. R() Écos Ê V [R()] d cos ˆ d sin c [ ( )]. R() Ê V [R()] d d Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

Chpter 6 Applictions of Definite Integrls È. R() Ê V [R()] d d; cu Ê du d; Ê u, Ê u d Ä V u du ( ) u. For the sketch given,, ; R(), r() È cos ; V [R()] [r()] d Î Î ( cos ) d ( cos ) d [ sin ] ˆ cî Î d. For the sketch given, c, d ; R(), r() tn ; V [R()] [r()] d c Î Î Î tn d sec d [ tn ] ˆ. r() nd R() Ê V [R()] [r()] d ˆ d 6. r() È nd R() Ê V [R()] [r()] d ( ) d ˆ 7. r() nd R() c Ê V [R()] [r()] d c ( ) d c c 69 d d c 68 d 6 8 ˆ 6 ˆ 8 ˆ 8 8 ˆ 8 6 7 Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

Section 6. Volumes Using Cross-Sections 8. r() nd R() c Ê V [R()] [r()] d c ( ) d c c 6 8 d d c 9 d 8 ˆ 8 ˆ ˆ 9. r() sec nd R() È Î Ê V [R()] [r()] d c Î Î c Î Î ˆ ˆ ( ) Î sec d [ tn ]. R() sec nd r() tn Ê V [R()] [r()] d sec tn d d []. r() nd R() Ê V [R()] [r()] d c( ) dd d d ˆ. R() nd r() Ê V [R()] [r()] d d ˆ c ( ) d d c d d Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

Chpter 6 Applictions of Definite Integrls. R() nd r() È Ê V [R()] [r()] d ( ) d (6 8) 8. R() È nd r() È È Ê V [R()] [r()] d È È c d d d È È. R() nd r() È Ê V [R()] [r()] d ˆ d È ˆ È d ˆ È d Î ˆ ˆ 8 8 7 6 6 Î 6. R() nd r() Ê V [R()] [r()] d ˆ d Î ˆ Î Î d ˆ Î Î d Î Î ˆ 7. () r() È nd R() Ê V [R()] [r()] d ( ) d (6 8) 8 () r() nd R() Ê V [R()] [r()] d d Î 8 6 6 8 (c) r() nd R() È Ê V [R()] [r()] d ˆ È d ˆ È d ˆ 6 Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

Section 6. Volumes Using Cross-Sections (d) r() nd R() Ê V [R()] [r()] d 6 d 8 6 6 6 8 d 8 d ˆ 8. () r() nd R() Ê V [R()] [r()] d ˆ Š 8 d d ˆ () r() nd R() Ê V [R()] [r()] d ˆ d Š d 8 8 Š d ˆ 6 ˆ 9. () r() nd R() c c c Ê V [R()] [r()] d d d ˆ ˆ c c d d c c ( ) Ê c c d d c c ( ) () r() nd R() Ê V [R()] [r()] d d ˆ (c) r() nd R() V [R()] [r()] d d ˆ. () r() nd R() h Ê V [R()] [r()] d h ˆ h h d Š h d h h ˆ h ˆ h h ˆ h h h h Š ˆ h h h h h h h h () r() nd R() Ê V [R()] [r()] d d d h h Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

6 Chpter 6 Applictions of Definite Integrls. R() È nd r() È c Ê V [R()] [r()] d c ˆ È ˆ È d È d È d c re of semicircle of rdius c. () A cross section hs rdius r È nd re r. The volume is d c d. dv dv dv dh dh dh dv dh dt dh dt dt dt Ah dt () Vh Ahdh, so Ah. Therefore Ah, so. dh units units For h, the re is ), so. dt ) sec ) sec hc hc c c. () R() È Ê V d h (h ) Š h h ( h) h h h h Š h h h dv dh h ( h) h h () Given dt. m /sec nd m, find dt. From prt (), V(h) h Ê dv h h Ê dv dv dh h( h) dh Ê dh. dh dt dh dt dt dt h ( ) ( )(6) m/sec.. Suppose the solid is produced revolving out the -is. Cst shdow of the solid on plne prllel to the -plne. Use n pproimtion such s the Trpezoid Rule, to estimte c R d d Œ. n k d k^. The cross section of solid right circulr clinder with cone removed is disk with rdius R from which disk of rdius h hs een removed. Thus its re is A R h R h. The cross section of the hemisphere is disk of rdius ÈR h. Therefore its re is A Š ÈR h R h. We cn see tht A A. The ltitudes of oth solids re R. Appling Cvlieris Principle we find Volume of Hemisphere (Volume of Clinder) (Volume of Cone) R R R R R. 6 6 6 6. R() È6 Ê V [R()] d 6 d 6 d 6 6 6 96 6 6 6 Š ˆ ˆ ˆ ˆ 6 6 cm. The plum o will weigh out W (8.) 9 gm, to the nerest grm. 7. R() È6 Ê V [R()] d 6 d 6 c7 c7 c6 c6 7 6 7 6 (6)( 7) Š (6)( 6) Š 6(6 7) cm 8 cm ( Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

Section 6. Volume Using Clindricl Shells 7 ˆ cos c c sin d ˆ cos c c sin d ˆ sin ˆ ˆ dv dc ˆ ˆ 8 ˆ ˆ. (See lso the ccompning grph.) 8. () R() kc sin k, so V [R()] d (c sin ) d c c sin sin d c c cos c c ( c ) ˆ c c. Let V(c) c c. We find the etreme vlues of V(c): (c ) Ê c is criticl point, nd V ; Evlute V t the endpoints: V() nd V() ( ). Now we see tht the functions solute minimum vlue is, tken on t the criticl point c () From the discussion in prt () we conclude tht the functions solute mimum vlue is the endpoint c. (c) The grph of the solids volume s function of c for Ÿ c Ÿ is given t the right. As c moves w from [ß ] the volume of the solid increses without ound. If we pproimte the solid s set of solid disks, we cn see tht the rdius of tpicl disk increses without ounds s c moves w from [ß ]., tken on t 9. Volume of the solid generted rotting the region ounded the -is nd f from to out the -is is V [f()] d, nd the volume of the solid generted rotting the sme region out the line f() d f() d is V [f() ] d 8. Thus [f() ] d [f()] d 8 Ê [f()] f() [f()] d Ê f() d Ê f() d d Ê Ê 6. Volume of the solid generted rotting the region ounded the -is nd f from to out the -is is V [f()] d 6, nd the volume of the solid generted rotting the sme region out the line is V [f() ] d. Thus [f() ] d [f()] d 6 Ê [f()] f() [f()] d Ê f() d Ê f() d d Ê f() d Ê f() d 6. VOLUME USING CYLINDRICAL SHELLS. For the sketch given,, ; V ˆ Š d Š d Š d ˆ 6 rdius height 6 6 6. For the sketch given,, ; V ˆ Š d Š d Š d ( ) 6 rdius height 6. For the sketch given, c, d È ; d È È È V ˆ d Š d d c rdius height Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

8 Chpter 6 Applictions of Definite Integrls. For the sketch given, c, d È ; d È È È V ˆ d Š c d d d c rdius height. For the sketch given,, È ; È V ˆ Š d Š È d; rdius height u Ê du d; Ê u, È Ê u Î Ä V u du Î u ˆ Î ˆ (8 ) 6. For the sketch given,, ; V ˆ 9 Š d Š d; rdius height È 9 cu 9 Ê du d Ê du 9 d; Ê u 9, Ê u 6d 6 Ä V Î u du 6 Î u Š È6 È9 6 9 * 9 7., ; V ˆ Š d ˆ d rdius height d d c d 8 8., ; V ˆ d ˆ Š d rdius height Š d d c d 9., ; V ˆ d Š c( ) dd rdius height d 6 ˆ ˆ., ; V ˆ d Š c d d rdius height d d ˆ Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

., ; V ˆ Š d È ( ) d rdius height ˆ Î d Î ˆ ˆ 7 Section 6. Volume Using Clindricl Shells 9., ; V ˆ d ˆ Î Š d rdius height Î Î Î d ˆ (8), Ÿ sin, Ÿ. () f() Ê f() ; since sin we hve,, sin, Ÿ f() Ê f() sin, Ÿ Ÿ sin, () V ˆ Š d f() d nd f() sin, Ÿ Ÿ prt () sin rdius height Ê V sin d [ cos ] ( cos cos ) tn, Ÿ tn, Ÿ /. () g() Ê g() ; since tn we hve,, g() tn, Ÿ / Ê g() tn, Ÿ Ÿ / tn, Î () V ˆ Š d g() d nd g() tn, Ÿ Ÿ / prt (). c, d ; rdius height Î Î Î Ê V tn d sec d [tn ] ˆ V ˆ Š d È ( ) d d c rdius height È È Î ˆ Î d Š È 8 8 Š 6Š Š È 6 Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

Chpter 6 Applictions of Definite Integrls 6. c, d ; d V ˆ d Š c ( ) dd c rdius height ˆ 6 d 6ˆ 6 7. c, d ; d V ˆ d Š d c rdius height ˆ 8 d ˆ 6 6 8. c, d ; d V ˆ d Š d c rdius height 6 d d ˆ 9. c, d ; d V ˆ Š d [ ( )]d c rdius height d c d. c, d ; V ˆ d ˆ Š d c rdius height d 8 d c d. c, d ; d V ˆ d Š c( ) dd c rdius height d ˆ (88) 8 6 6 6 Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

Section 6. Volume Using Clindricl Shells. c, d ; d V ˆ d Š c( ) d d c rdius height d 6 6 ˆ (). () V ˆ Š d d 6 d c d 6 rdius height rdius height () V ˆ d d 6 d 6 Š 6ˆ 8 8 rdius height (c) V ˆ d d 6 d 6 Š 6ˆ 8 8 d 6 6 6 c rdius height 9 (d) V ˆ d ˆ d ˆ d Š 6 d 6 6 6 c rdius height 6 9 (e) V ˆ d 7 ˆ d ˆ d Š 878 6 d 6 6 6 c rdius height 9 (f) V ˆ d ˆ d ˆ d Š 8 96 rdius height. () V ˆ d 8 d 8 d Š ˆ 6 () V ˆ Š d 8 d 8 d rdius height ˆ 6 ˆ Š rdius height ˆ 6 d 8 8 ˆ Î Î 6 Š 7Î 8 6 768 c rdius height 7 7 7 8 6 (c) V d 8 d 6 8 d 6 6 8 (d) V d d d 8 d 8 8 8 c rdius height 7 8 76 7 7 d 8 8 ˆ Î ˆ Î Î Š 7Î Î 8 c rdius height 7 8 96 7 7 (e) V ˆ Î d 8 d ˆ Î Î Š 8 d Î 7Î 6 ˆ 96 (f) V d d d ˆ rdius height c c 8 8 ˆ 7 ˆ Š rdius height c c 6 ˆ 7 d ˆ Š ˆ È ˆ È ˆ È c rdius height Î d 8 ˆ Î d Î Î 8 6 6 6 7 ˆ ˆ. () V ˆ d d d Š. () V d d d (c) V d d d Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

Chpter 6 Applictions of Definite Integrls d (d) V ˆ Š d ˆ È ˆ È d ˆ È d c rdius height ˆ È Î ˆ Î È 8 Î Î Î 8 Î ˆ 8 ˆ 6 6 6 ˆ 8 8 d 6 8 d 8 8 8 6. () V ˆ Š d d d rdius height c c 6 ˆ ˆ 6 6 6 6 d ˆ Š ˆ È ˆ È É Š É c rdius height () V d d d Î d È d cu Ê u Ê du d; Ê u, Ê u d È 6 9 6 6 8 9 Î uèu du 9 ˆ Èu u Î du 9 u Î u Î È È È È Š 8È È 6 8 6 88 87 9 9 d 7. () V ˆ Š d d d c rdius height 6 d ˆ Š c d c rdius height d ˆ ˆ d ˆ ˆ 8 ˆ 8 c rdius height ˆ 8 8 ˆ 8 6 d ˆ ˆ Š c d ˆ c rdius height ˆ () V d ( ) d ( ) d (c) V Š d c dd d d ( 9 ) (d) V d d d ˆ d ˆ d ˆ (8 9 ) 6 d 8. () V ˆ Š d Š d Š d Š d c rdius height Š ˆ ˆ ˆ 8 6 d ˆ Š Š Š c rdius height Š ˆ 6 6 6 8 d ˆ Š Š Š c rdius height Š ˆ 6 6 6 d ˆ ˆ Š Š ˆ Š c rdius height 8 8 Š ˆ 6 6 6 8 6 6 () V d ( ) d ( ) d d (c) V d ( ) d ( ) d d 8 (d) V d d d d Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

Section 6. Volume Using Clindricl Shells rdius height 9. () Aout -is: V cd ˆ Š d ˆ È d ˆ Î d ˆ Î rdius height Aout -is: V ˆ Š d ˆ d d () Aout -is: R nd r V R r Ê d c dd ˆ Aout -is: R È nd r V R r Ê d c dd ˆ. () V R r d ˆ d ˆ d () V ˆ d ˆ Š d rdius height ˆ d Š d d c ˆ ˆ ˆ ˆ rdius height ˆ ˆ Š ˆ d ) ) ( ) (c) V Š d d d Š ) d ) (d) V R r d ) d d. () V ˆ Š d ( ) d d c rdius height d ˆ 8 ˆ ˆ () 7 () V ˆ Š d ( ) d d rdius height 8 8 ˆ ˆ ˆ 6 Š rdius height 8 ˆ 8 ˆ 8 ˆ d ˆ () Š c rdius height ˆ ˆ ˆ ˆ ˆ (c) V d ( ) d d (d) V d ( )( ) d ( ) Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

Chpter 6 Applictions of Definite Integrls. () V ˆ Š d d d c rdius height d Š 8 rdius height () V ˆ Š d ˆ È d ˆ Î d Š 6 Î 6 Š È rdius height ˆ 6 (8 6) (c) V ˆ d ( ) ˆ d ˆ Î Î 8 d 8 ˆ 6 ( 9) () 8 Î Î 6 6 d ˆ Š c rdius height 6 6 8 (d) V d ( ) d d ˆ (). () V ˆ Š d d d c rdius height d ˆ rdius height () V cd ˆ Š d () d d ˆ () 7 6 rdius height. () V cd ˆ Š d c dd d ˆ ( 6) () Use the wsher method: d c 97 7 7 ( V cr () r () dd d d ˆ ( ) (c) Use the wsher method: d c ( d 7 7 (7 ) d ˆ Š c d c rdius height d d ˆ 6 ( ) V cr () r () dd c d d d ˆ (d) V d ( ) d ( ) d Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

. () V ˆ Š d ˆ È 8 d d c rdius height Š È Î d È Î È Š È Š 8 8 ˆ (8) Section 6. Volume Using Clindricl Shells () V ˆ Š d Š È Î Î d Š d rdius height 8 8 8 6 6 Š Š ) ( ( ) * rdius height 6. () V ˆ Š d c d d 6 d d ˆ () V ˆ Š d ( ) c dd ( ) d rdius height ˆ 6 d (68) 7. () V cr () r () dd ˆ Î d Î6 Î ( ) ˆ Î 6 ˆ 7 9 6 6 rdius height 6 ˆ d 6 ˆ ˆ ˆ 8 8 (8 ) 9 6 () V ˆ Š d Š d 8. () V cr () r () dd Š d d c 6 ˆ ˆ 6 8 6 ( 6 6 ) 8 8 ˆ Š Š rdius height Î È ˆ Î Î d Î Î ˆ ˆ ˆ 8 6 6 8 ( 6 8 8 ) 8 () V d d 9. () H=: V V V V [R ()] d nd V [R ()] with R () É nd R () È,, ;, Ê two integrls re required Copright Person Eduction, Inc. Pulishing s Addison-Wesle.

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