6.2 Volumes. V Ah 430 CHAPTER 6 APPLICATIONS OF INTEGRATION. h h w l (c) Rectangular box V=lwh. (a) Cylinder V=Ah. (b) Circular cylinder

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1 43 CHAPTER 6 APPLICATIONS OF INTEGRATION 6. Volumes In tring to find the volume of solid we fce the sme tpe of problem s in finding res. We hve n intuitive ide of wht volume mens, but we must mke this ide precise b using clculus to give n ect definition of volume. We strt with simple tpe of solid clled clinder (or, more precisel, right clinder). As illustrted in Figure (), clinder is bounded b plne region B, clled the bse, nd congruent region B in prllel plne. The clinder consists of ll points on line segments tht re perpendiculr to the bse nd join B to B. If the re of the bse is A nd the height of the clinder (the distnce from B to B ) is h, then the volume V of the clinder is defined s V Ah In prticulr, if the bse is circle with rdius r, then the clinder is circulr clinder with volume V r h [see Figure (b)], nd if the bse is rectngle with length l nd width w, then the clinder is rectngulr bo (lso clled rectngulr prllelepiped) with volume V lwh [see Figure (c)]. FIGURE B h B () Clinder V=Ah (b) Circulr clinder V=πr@h r h h w l (c) Rectngulr bo V=lwh For solid S tht isn t clinder we first cut S into pieces nd pproimte ech piece b clinder. We estimte the volume of S b dding the volumes of the clinders. We rrive t the ect volume of S through limiting process in which the number of pieces becomes lrge. We strt b intersecting S with plne nd obtining plne region tht is clled crosssection of S. Let A be the re of the cross-section of S in plne P perpendiculr to the -is nd pssing through the point, where b. (See Figure. Think of slicing S with knife through nd computing the re of this slice.) The cross-sectionl re A will vr s increses from to b. P A() A(b) FIGURE b

2 SECTION 6. VOLUMES 43 Let s divide S into n slbs of equl width b using the plnes P, P,...to slice the solid. (Think of slicing lof of bred.) If we choose smple points * i in i, i, we cn pproimte the ith slb S i (the prt of S tht lies between the plnes P i nd P i ) b clinder with bse re A * i nd height. (See Figure 3.) Î S * b i i- i = ß =b FIGURE 3 The volume of this clinder is A * i, so n pproimtion to our intuitive conception of the volume of the ith slb is S i V S i A i * Adding the volumes of these slbs, we get n pproimtion to the totl volume (tht is, wht we think of intuitivel s the volume): V n i A i * It cn be proved tht this definition is independent of how S is situted with respect to the -is. In other words, no mtter how we slice S with prllel plnes, we lws get the sme nswer for V. This pproimtion ppers to become better nd better s n l. (Think of the slices s becoming thinner nd thinner.) Therefore we define the volume s the limit of these sums s n l. But we recognize the limit of Riemnn sums s definite integrl nd so we hve the following definition. Definition of Volume Let S be solid tht lies between nd b. If the cross-sectionl re of S in the plne P, through nd perpendiculr to the -is, is A, where A is continuous function, then the volume of S is V lim n l n i A * i b A d _r r When we use the volume formul V b A d, it is importnt to remember tht A is the re of moving cross-section obtined b slicing through perpendiculr to the -is. Notice tht, for clinder, the cross-sectionl re is constnt: A A for ll. So our definition of volume gives V b Ad A b ; this grees with the formul V Ah. EXAMPLE Show tht the volume of sphere of rdius r is V 4 3 r 3. FIGURE 4 SOLUTION If we plce the sphere so tht its center is t the origin (see Figure 4), then the plne intersects the sphere in circle whose rdius (from the Pthgoren Theorem) P

3 43 CHAPTER 6 APPLICATIONS OF INTEGRATION is sr. So the cross-sectionl re is A r Using the definition of volume with r nd b r, we hve V r A d r r d r r r r d r r 3 3 r r 3 r 3 3 (The integrnd is even.) Figure 5 illustrtes the definition of volume when the solid is sphere with rdius 4 r. From the result of Emple, we know tht the volume of the sphere is 3, which is pproimtel Here the slbs re circulr clinders, or disks, nd the three prts of Figure 5 show the geometric interprettions of the Riemnn sums TEC Visul 6.A shows n nimtion of Figure 5. n A i n i i i when n 5,, nd if we choose the smple points * i to be the midpoints i. Notice tht s we increse the number of pproimting clinders, the corresponding Riemnn sums become closer to the true volume. () Using 5 disks, VÅ4.76 (b) Using disks, VÅ4.97 (c) Using disks, VÅ4.94 FIGURE 5 Approimting the volume of sphere with rdius v EXAMPLE Find the volume of the solid obtined b rotting bout the -is the region under the curve s from to. Illustrte the definition of volume b sketching tpicl pproimting clinder. SOLUTION The region is shown in Figure 6(). If we rotte bout the -is, we get the solid shown in Figure 6(b). When we slice through the point, we get disk with rdius s. The re of this cross-section is A (s ) nd the volume of the pproimting clinder ( disk with thickness ) is A

4 SECTION 6. VOLUMES 433 The solid lies between nd, so its volume is V A d d =œ Did we get resonble nswer in Emple? As check on our work, let s replce the given region b squre with bse, nd height. If we rotte this squre, we get clinder with rdius, height, nd volume. We computed tht the given solid hs hlf this volume. Tht seems bout right. œ Î FIGURE 6 () (b) v EXAMPLE 3 Find the volume of the solid obtined b rotting the region bounded b 3, 8, nd bout the -is. SOLUTION The region is shown in Figure 7() nd the resulting solid is shown in Figure 7(b). Becuse the region is rotted bout the -is, it mkes sense to slice the solid perpendiculr to the -is nd therefore to integrte with respect to. If we slice t height, we get circulr disk with rdius, where s 3. So the re of crosssection through is A (s 3 ) 3 nd the volume of the pproimting clinder pictured in Figure 7(b) is A 3 Since the solid lies between nd 8, its volume is V 8 A d 8 3 d [ ] =8 8 = = or 3 =œ Î (, ) FIGURE 7 () (b)

5 434 CHAPTER 6 APPLICATIONS OF INTEGRATION TEC Visul 6.B shows how solids of revolution re formed. EXAMPLE 4 The region enclosed b the curves nd is rotted bout the -is. Find the volume of the resulting solid. SOLUTION The curves nd intersect t the points, nd,. The region between them, the solid of rottion, nd cross-section perpendiculr to the -is re shown in Figure 8. A cross-section in the plne P hs the shpe of wsher (n nnulr ring) with inner rdius nd outer rdius, so we find the cross-sectionl re b subtrcting the re of the inner circle from the re of the outer circle: Therefore we hve A 4 V A d 4 d = (, ) = A() (, ) FIGURE 8 () (b) (c) EXAMPLE 5 Find the volume of the solid obtined b rotting the region in Emple 4 bout the line. SOLUTION The solid nd cross-section re shown in Figure 9. Agin the cross-section is wsher, but this time the inner rdius is nd the outer rdius is. 4 = = - - FIGURE 9 = =

6 SECTION 6. VOLUMES 435 The cross-sectionl re is nd so the volume of S is The solids in Emples 5 re ll clled solids of revolution becuse the re obtined b revolving region bout line. In generl, we clculte the volume of solid of revolution b using the bsic defining formul nd we find the cross-sectionl re A or A in one of the following ws: N A V A d d d V b A d or V d A d If the cross-section is disk (s in Emples 3), we find the rdius of the disk (in terms of or ) nd use A rdius c N If the cross-section is wsher (s in Emples 4 nd 5), we find the inner rdius r in nd outer rdius r out from sketch (s in Figures 8, 9, nd ) nd compute the re of the wsher b subtrcting the re of the inner disk from the re of the outer disk: A outer rdius inner rdius r in r out FIGURE The net emple gives further illustrtion of the procedure.

7 436 CHAPTER 6 APPLICATIONS OF INTEGRATION EXAMPLE 6 Find the volume of the solid obtined b rotting the region in Emple 4 bout the line. SOLUTION Figure shows horizontl cross-section. It is wsher with inner rdius nd outer rdius s, so the cross-sectionl re is The volume is A outer rdius inner rdius V A d ( s ) [( s ) ] d 3 4 (s ) d œ + = =œ FIGURE =_ We now find the volumes of two solids tht re not solids of revolution. TEC Visul 6.C shows how the solid in Figure is generted. EXAMPLE 7 Figure shows solid with circulr bse of rdius. Prllel crosssections perpendiculr to the bse re equilterl tringles. Find the volume of the solid. SOLUTION Let s tke the circle to be. The solid, its bse, nd tpicl crosssection t distnce from the origin re shown in Figure 3. C =œ B(, ) C B _ A A A œ œ3 6 6 B () The solid (b) Its bse (c) A cross-section FIGURE Computer-generted picture of the solid in Emple 7 FIGURE 3 is Since B lies on the circle, we hve s nd so the bse of the tringle ABC s. Since the tringle is equilterl, we see from Figure 3(c) tht its AB

8 SECTION 6. VOLUMES 437 height is s3 s3 s. The cross-sectionl re is therefore A s s3 s s3 nd the volume of the solid is V A d s3 d s3 d s3 3 3 v EXAMPLE 8 Find the volume of prmid whose bse is squre with side L nd whose height is h. SOLUTION We plce the origin O t the verte of the prmid nd the -is long its centrl is s in Figure 4. An plne P tht psses through nd is perpendiculr to the -is intersects the prmid in squre with side of length s, s. We cn epress s in terms of b observing from the similr tringles in Figure 5 tht s h L s L nd so s L h. [Another method is to observe tht the line OP hs slope L h nd so its eqution is L h.] Thus the cross-sectionl re is 4s3 3 A s L h P O h O s L h FIGURE 4 FIGURE 5 h FIGURE 6 The prmid lies between nd h, so its volume is V h A d h L h d L h 3 3 NOTE We didn t need to plce the verte of the prmid t the origin in Emple 8. We did so merel to mke the equtions simple. If, insted, we hd plced the center of the bse t the origin nd the verte on the positive -is, s in Figure 6, ou cn verif tht we would hve obtined the integrl V h L h h d L h 3 h L h 3

9 438 CHAPTER 6 APPLICATIONS OF INTEGRATION EXAMPLE 9 A wedge is cut out of circulr clinder of rdius 4 b two plnes. One plne is perpendiculr to the is of the clinder. The other intersects the first t n ngle of 3 long dimeter of the clinder. Find the volume of the wedge. A 4 C B =œ 6- SOLUTION If we plce the -is long the dimeter where the plnes meet, then the bse of the solid is semicircle with eqution s6, 4 4. A crosssection perpendiculr to the -is t distnce from the origin is tringle ABC, s shown in Figure 7, whose bse is s6 nd whose height is. Thus the cross-sectionl re is BC tn 3 s6 s3 A s6 s3 6 s6 s3 C nd the volume is V 4 A d s3 d A 3 B s3 4 6 d s FIGURE 7 8 3s3 For nother method see Eercise Eercises 8 Find the volume of the solid obtined b rotting the region bounded b the given curves bout the specified line. Sketch the region, the solid, nd tpicl disk or wsher..,,, ; bout the -is., ; bout the -is 3. s,, 5; bout the -is 4. s5,,, 4; bout the -is 5. s,, 9; bout the -is 6. ln,,, ; bout the -is 7. 3,, ; bout the -is 8. 4, 5 ; bout the -is 9., ; bout the -is 4. sin, cos, 4; bout 5. 3,, ; bout 6.,,, ; bout 7., ; bout 3 8.,,, 4; bout 9 3 Refer to the figure nd find the volume generted b rotting the given region bout the specified line. C(, ) T =œ $ T T B(, ). 4,, ; bout the -is., ; bout O A(, ). e,, ; bout 3. sec, 3; bout 9. bout OA.. bout AB. bout OC bout BC ; Grphing clcultor or computer required CAS Computer lgebr sstem required. Homework Hints vilble t stewrtclculus.com

10 SECTION 6. VOLUMES bout OA 4. bout OC 5. bout AB 6. bout BC 7. 3 bout OA 8. 3 bout OC 9. 3 bout AB 3. 3 bout BC 3 34 Set up n integrl for the volume of the solid obtined b rotting the region bounded b the given curves bout the specified line. Then use our clcultor to evlute the integrl correct to five deciml plces. 3. e,,, () About the -is (b) About 3., cos, () About the -is (b) About () About (b) About 34.,, () About the -is (b) About the -is 44. A log m long is cut t -meter intervls nd its crosssectionl res A (t distnce from the end of the log) re listed in the tble. Use the Midpoint Rule with n 5 to estimte the volume of the log. 45. () If the region shown in the figure is rotted bout the -is to form solid, use the Midpoint Rule with n 4 to estimte the volume of the solid. 4 (m) A ( m ) (m) A ( ) m ; Use grph to find pproimte -coordintes of the points of intersection of the given curves. Then use our clcultor to find (pproi mtel) the volume of the solid obtined b rotting bout the -is the region bounded b these curves. CAS 35. cos, sin, Use computer lgebr sstem to find the ect volume of the solid obtined b rotting the region bounded b the given curves bout the specified line. 37. sin,, ; bout 38., e ; bout Ech integrl represents the volume of solid. Describe the solid. 39. sin d d 4 e e 4. cos d d 43. A CAT scn produces equll spced cross-sectionl views of humn orgn tht provide informtion bout the orgn otherwise obtined onl b surger. Suppose tht CAT scn of humn liver shows cross-sections spced.5 cm prt. The liver is 5 cm long nd the cross-sectionl res, in squre centimeters, re, 8, 58, 79, 94, 6, 7, 8, 63, 39, nd. Use the Midpoint Rule to estimte the volume of the liver. CAS (b) Estimte the volume if the region is rotted bout the -is. Agin use the Midpoint Rule with n () A model for the shpe of bird s egg is obtined b rotting bout the -is the region under the grph of f 3 b c d s Use CAS to find the volume of such n egg. (b) For Red-throted Loon,.6, b.4, c., nd d.54. Grph f nd find the volume of n egg of this species Find the volume of the described solid S. 47. A right circulr cone with height h nd bse rdius r 48. A frustum of right circulr cone with height h, lower bse rdius R, nd top rdius r r h R 49. A cp of sphere with rdius r nd height h h r

11 44 CHAPTER 6 APPLICATIONS OF INTEGRATION 5. A frustum of prmid with squre bse of side b, squre top of side, nd height h 6. () Set up n integrl for the volume of solid torus (the donut-shped solid shown in the figure) with rdii r nd R. (b) B interpreting the integrl s n re, find the volume of the torus. b Wht hppens if b? Wht hppens if? R r 5. A prmid with height h nd rectngulr bse with dimensions b nd b 5. A prmid with height h nd bse n equilterl tringle with side ( tetrhedron) 6. Solve Emple 9 tking cross-sections to be prllel to the line of intersection of the two plnes. 63. () Cvlieri s Principle sttes tht if fmil of prllel plnes gives equl cross-sectionl res for two solids S nd S, then the volumes of S nd S re equl. Prove this principle. (b) Use Cvlieri s Principle to find the volume of the oblique clinder shown in the figure. 53. A tetrhedron with three mutull perpendiculr fces nd three mutull perpendiculr edges with lengths 3 cm, 4 cm, nd 5 cm r h 54. The bse of S is circulr disk with rdius r. Prllel crosssections perpendiculr to the bse re squres. 64. Find the volume common to two circulr clinders, ech with rdius r, if the es of the clinders intersect t right ngles. 55. The bse of S is n ellipticl region with boundr curve Cross-sections perpendiculr to the -is re isosceles right tringles with hpotenuse in the bse. 56. The bse of S is the tringulr region with vertices,,,, nd,. Cross-sections perpendiculr to the -is re equilterl tringles. 57. The bse of S is the sme bse s in Eercise 56, but crosssections perpendiculr to the -is re squres. 58. The bse of S is the region enclosed b the prbol nd the -is. Cross-sections perpendiculr to the -is re squres. 59. The bse of S is the sme bse s in Eercise 58, but crosssections perpendiculr to the -is re isosceles tringles with height equl to the bse. 6. The bse of S is circulr disk with rdius r. Prllel crosssections perpendiculr to the bse re isosceles tringles with height h nd unequl side in the bse. () Set up n integrl for the volume of S. (b) B interpreting the integrl s n re, find the volume of S. 65. Find the volume common to two spheres, ech with rdius r, if the center of ech sphere lies on the surfce of the other sphere. 66. A bowl is shped like hemisphere with dimeter 3 cm. A hev bll with dimeter cm is plced in the bowl nd wter is poured into the bowl to depth of h centimeters. Find the volume of wter in the bowl. 67. A hole of rdius r is bored through the middle of clinder of rdius R r t right ngles to the is of the clinder. Set up, but do not evlute, n integrl for the volume cut out. 68. A hole of rdius r is bored through the center of sphere of rdius R r. Find the volume of the remining portion of the sphere. 69. Some of the pioneers of clculus, such s Kepler nd Newton, were inspired b the problem of finding the volumes of wine

12 SECTION 6.3 VOLUMES BY CYLINDRICAL SHELLS 44 brrels. (In fct Kepler published book Stereometri doliorum in 65 devoted to methods for finding the volumes of brrels.) The often pproimted the shpe of the sides b prbols. () A brrel with height h nd mimum rdius R is constructed b rotting bout the -is the prbol R c, h h, where c is positive constnt. Show tht the rdius of ech end of the brrel is r R d, where d ch 4. (b) Show tht the volume enclosed b the brrel is V 3 h(r r 5 d ) 7. Suppose tht region hs re A nd lies bove the -is. When is rotted bout the -is, it sweeps out solid with volume V. When is rotted bout the line k (where k is positive number), it sweeps out solid with volume V. Epress V in terms of, k, nd A. V 6.3 Volumes b Clindricl Shells FIGURE L =? = - R =? Some volume problems re ver difficult to hndle b the methods of the preceding section. For instnce, let s consider the problem of finding the volume of the solid obtined b rotting bout the -is the region bounded b 3 nd. (See Figure.) If we slice perpendiculr to the -is, we get wsher. But to compute the inner rdius nd the outer rdius of the wsher, we d hve to solve the cubic eqution 3 for in terms of ; tht s not es. Fortuntel, there is method, clled the method of clindricl shells, tht is esier to use in such cse. Figure shows clindricl shell with inner rdius r, outer rdius r, nd height h. Its volume V is clculted b subtrcting the volume V of the inner clinder from the volume V of the outer clinder: V V V r r r Îr r h r h r r h r r r r h h r r h r r If we let r r (the thickness of the shell) nd r r r r (the verge rdius of the shell), then this formul for the volume of clindricl shell becomes FIGURE nd it cn be remembered s V rh r V [circumference][height][thickness] Now let S be the solid obtined b rotting bout the -is the region bounded b f [where f ],,, nd b, where b. (See Figure 3.) =ƒ =ƒ b b FIGURE 3

13 44 CHAPTER 6 APPLICATIONS OF INTEGRATION We divide the intervl, b into n subintervls i, i of equl width nd let i be the midpoint of the ith subintervl. If the rectngle with bse i, i nd height f i is rotted bout the -is, then the result is clindricl shell with verge rdius i, height f i, nd thickness (see Figure 4), so b Formul its volume is V i i f i =ƒ =ƒ =ƒ b b b i- i i FIGURE 4 Therefore n pproimtion to the volume V of S is given b the sum of the volumes of these shells: V n V i n i f i This pproimtion ppers to become better s n l. But, from the definition of n integrl, we know tht lim n l n i i Thus the following ppers plusible: i i f i b f d The volume of the solid in Figure 3, obtined b rotting bout the -is the region under the curve f from to b, is V b f d where b The rgument using clindricl shells mkes Formul seem resonble, but lter we will be ble to prove it (see Eercise 7 in Section 7.). The best w to remember Formul is to think of tpicl shell, cut nd flttened s in Figure 5, with rdius, circumference, height f, nd thickness or d: b f d circumference height thickness ƒ ƒ π Î FIGURE 5

14 SECTION 6.3 VOLUMES BY CYLINDRICAL SHELLS 443 This tpe of resoning will be helpful in other situtions, such s when we rotte bout lines other thn the -is. EXAMPLE Find the volume of the solid obtined b rotting bout the -is the region bounded b 3 nd. SOLUTION From the sketch in Figure 6 we see tht tpicl shell hs rdius, circumference, nd height f 3. So, b the shell method, the volume is - V 3 d 3 4 d [ ] (8 3 5 ) 6 5 FIGURE 6 It cn be verified tht the shell method gives the sme nswer s slicing. Figure 7 shows computer-generted picture of the solid whose volume we computed in Emple. FIGURE 7 NOTE Compring the solution of Emple with the remrks t the beginning of this section, we see tht the method of clindricl shells is much esier thn the wsher method for this problem. We did not hve to find the coordintes of the locl mimum nd we did not hve to solve the eqution of the curve for in terms of. However, in other emples the methods of the preceding section m be esier. = = shell height=- v EXAMPLE Find the volume of the solid obtined b rotting bout the -is the region between nd. SOLUTION The region nd tpicl shell re shown in Figure 8. We see tht the shell hs rdius, circumference, nd height. So the volume is V d 3 d FIGURE As the following emple shows, the shell method works just s well if we rotte bout the -is. We simpl hve to drw digrm to identif the rdius nd height of shell. v EXAMPLE 3 Use clindricl shells to find the volume of the solid obtined b rotting bout the -is the region under the curve s from to.

15 444 CHAPTER 6 APPLICATIONS OF INTEGRATION = FIGURE 9 shell height=- shell rdius= = SOLUTION This problem ws solved using disks in Emple in Section 6.. To use shells we relbel the curve s (in the figure in tht emple) s in Figure 9. For rottion bout the -is we see tht tpicl shell hs rdius, circumference, nd height. So the volume is V d 3 d In this problem the disk method ws simpler. 4 4 v EXAMPLE 4 Find the volume of the solid obtined b rotting the region bounded b nd bout the line. SOLUTION Figure shows the region nd clindricl shell formed b rottion bout the line. It hs rdius, circumference, nd height. =- = FIGURE The volume of the given solid is V d 3 3 d Eercises. Let S be the solid obtined b rotting the region shown in. Let S be the solid obtined b rotting the region shown in the the figure bout the -is. Eplin wh it is wkwrd to use figure bout the -is. Sketch tpicl clindricl shell nd slicing to find the volume V of S. Sketch tpicl pproi - find its circumference nd height. Use shells to find the volume mting shell. Wht re its circumference nd height? Use shells of S. Do ou think this method is preferble to slicing? Eplin. to find V. =(-)@ =sin{ } œ π ; Grphing clcultor or computer required CAS Computer lgebr sstem required. Homework Hints vilble t stewrtclculus.com

16 SECTION 6.3 VOLUMES BY CYLINDRICAL SHELLS Use the method of clindricl shells to find the volume generted b rotting the region bounded b the given curves bout the -is. 3. s 3,, 4. 3,,, 5. e,,, 6. 4, 7., 6 8. Let V be the volume of the solid obtined b rotting bout the -is the region bounded b s nd. Find V both b slicing nd b clindricl shells. In both cses drw digrm to eplin our method. 9 4 Use the method of clindricl shells to find the volume of the solid obtined b rotting the region bounded b the given curves bout the -is. 9.,,,. s,,. 3, 8,. 4 3, 3., , 4 5 Use the method of clindricl shells to find the volume generted b rotting the region bounded b the given curves bout the specified is. 5. 4,, ; bout 6. s,, ; bout 7. 4, 3; bout 8., ; bout 9. 3,, ; bout., ; bout 6 () Set up n integrl for the volume of the solid obtined b rotting the region bounded b the given curve bout the specified is. (b) Use our clcultor to evlute the integrl correct to five deciml plces.. e,, ; bout the -is. tn,, 4; bout 3. cos 4, cos 4, ; bout 4., 3 ; bout 5. ssin,, ; bout , 4; bout 5 7. Use the Midpoint Rule with n 5 to estimte the volume obtined b rotting bout the -is the region under the curve s 3,. 8. If the region shown in the figure is rotted bout the -is to form solid, use the Midpoint Rule with n 5 to estimte the volume of the solid. 9 3 Ech integrl represents the volume of solid. Describe the solid ; Use grph to estimte the -coordintes of the points of intersection of the given curves. Then use this informtion nd our clcultor to estimte the volume of the solid obtined b rotting bout the -is the region enclosed b these curves. CAS d d 3 d 4 cos sin d 33. e, s 34. 3, Use computer lgebr sstem to find the ect volume of the solid obtined b rotting the region bounded b the given curves bout the specified line. 35. sin, sin 4, ; bout sin,, ; bout The region bounded b the given curves is rotted bout the specified is. Find the volume of the resulting solid b n method , ; bout the -is , ; bout the -is 39., ; bout the -is 4., ; bout the -is

17 446 CHAPTER 6 APPLICATIONS OF INTEGRATION 4. ; bout the -is 4. 3, 4; bout 43., ; bout 44. Let T be the tringulr region with vertices,,,, nd,, nd let V be the volume of the solid generted when T is rotted bout the line, where. Epress in terms of V Use clindricl shells to find the volume of the solid. 45. A sphere of rdius r 46. The solid torus of Eercise 6 in Section A right circulr cone with height h nd bse rdius r 48. Suppose ou mke npkin rings b drilling holes with different dimeters through two wooden blls (which lso hve different dimeters). You discover tht both npkin rings hve the sme height h, s shown in the figure. () Guess which ring hs more wood in it. (b) Check our guess: Use clindricl shells to compute the volume of npkin ring creted b drilling hole with rdius r through the center of sphere of rdius R nd epress the nswer in terms of h. h 6.4 Work The term work is used in everd lnguge to men the totl mount of effort required to perform tsk. In phsics it hs technicl mening tht depends on the ide of force. Intuitivel, ou cn think of force s describing push or pull on n object for emple, horizontl push of book cross tble or the downwrd pull of the erth s grvit on bll. In generl, if n object moves long stright line with position function s t, then the force F on the object (in the sme direction) is given b Newton s Second Lw of Motion s the product of its mss m nd its ccelertion: F m d s dt In the SI metric sstem, the mss is mesured in kilogrms (kg), the displcement in meters (m), the time in seconds (s), nd the force in newtons ( N kg m s ). Thus force of N cting on mss of kg produces n ccelertion of m s. In the US Customr sstem the fundmentl unit is chosen to be the unit of force, which is the pound. In the cse of constnt ccelertion, the force F is lso constnt nd the work done is defined to be the product of the force F nd the distnce d tht the object moves: W Fd work force distnce If F is mesured in newtons nd d in meters, then the unit for W is newton-meter, which is clled joule (J). If F is mesured in pounds nd d in feet, then the unit for W is footpound (ft-lb), which is bout.36 J. v EXAMPLE () How much work is done in lifting.-kg book off the floor to put it on desk tht is.7 m high? Use the fct tht the ccelertion due to grvit is t 9.8 m s. (b) How much work is done in lifting -lb weight 6 ft off the ground? SOLUTION () The force eerted is equl nd opposite to tht eerted b grvit, so Eqution gives F mt N

18 SECTION 6.4 WORK 447 nd then Eqution gives the work done s W Fd J (b) Here the force is given s F lb, so the work done is W Fd 6 ft-lb Notice tht in prt (b), unlike prt (), we did not hve to multipl b t becuse we were given the weight (which is force) nd not the mss of the object. Eqution defines work s long s the force is constnt, but wht hppens if the force is vrible? Let s suppose tht the object moves long the -is in the positive direction, from to b, nd t ech point between nd b force f cts on the object, where f is continuous function. We divide the intervl, b into n subintervls with endpoints,,..., n nd equl width. We choose smple point * i in the ith subintervl i, i. Then the force t tht point is f * i. If n is lrge, then is smll, nd since f is continuous, the vlues of f don t chnge ver much over the intervl i, i. In other words, f is lmost constnt on the intervl nd so the work W i tht is done in moving the prticle from to is pproimtel given b Eqution : i i W i f * i Thus we cn pproimte the totl work b 3 W n i f i * It seems tht this pproimtion becomes better s we mke n lrger. Therefore we define the work done in moving the object from to b s the limit of this quntit s n l. Since the right side of 3 is Riemnn sum, we recognize its limit s being definite integrl nd so 4 W lim n l n i f * i b f d frictionless surfce () Nturl position of spring FIGURE Hooke s Lw ƒ=k (b) Stretched position of spring EXAMPLE When prticle is locted distnce feet from the origin, force of pounds cts on it. How much work is done in moving it from to 3? SOLUTION The work done is 6 3 W 3 d 3 3 ft-lb. In the net emple we use lw from phsics: Hooke s Lw sttes tht the force required to mintin spring stretched units beond its nturl length is proportionl to : f k where k is positive constnt (clled the spring constnt). Hooke s Lw holds provided tht is not too lrge (see Figure )

19 448 CHAPTER 6 APPLICATIONS OF INTEGRATION v EXAMPLE 3 A force of 4 N is required to hold spring tht hs been stretched from its nturl length of cm to length of 5 cm. How much work is done in stretching the spring from 5 cm to 8 cm? SOLUTION According to Hooke s Lw, the force required to hold the spring stretched meters beond its nturl length is f k. When the spring is stretched from cm to 5 cm, the mount stretched is 5 cm.5 m. This mens tht f.5 4, so.5k 4 k Thus f 8 nd the work done in stretching the spring from 5 cm to 8 cm is * i FIGURE Î W.8 8d J v EXAMPLE 4 A -lb cble is ft long nd hngs verticll from the top of tll building. How much work is required to lift the cble to the top of the building? SOLUTION Here we don t hve formul for the force function, but we cn use n rgument similr to the one tht led to Definition 4. Let s plce the origin t the top of the building nd the -is pointing downwrd s in Figure. We divide the cble into smll prts with length. If * i is point in the ith such intervl, then ll points in the intervl re lifted b pproimtel the sme mount, nmel * i. The cble weighs pounds per foot, so the weight of the ith prt is. Thus the work done on the ith prt, in foot-pounds, is * i * i force distnce.8 If we hd plced the origin t the bottom of the cble nd the -is upwrd, we would hve gotten W d which gives the sme nswer. We get the totl work done b dding ll these pproimtions nd letting the number of prts become lrge (so l ): W lim n l n i * i d ], ft-lb EXAMPLE 5 A tnk hs the shpe of n inverted circulr cone with height m nd bse rdius 4 m. It is filled with wter to height of 8 m. Find the work required to empt the tnk b pumping ll of the wter to the top of the tnk. (The densit of wter is 3 kg m.) SOLUTION Let s mesure depths from the top of the tnk b introducing verticl coordinte line s in Figure 3. The wter etends from depth of m to depth of m nd so we divide the intervl, into n subintervls with endpoints,,..., n nd choose * i in the ith subintervl. This divides the wter into n lers. The ith ler is pproimted b circulr clinder with rdius r i nd height. We cn compute r i from similr tringles, using Figure 4, s follows: r i i * 4 r i 5 i *

20 SECTION 6.4 WORK 449 4m Thus n pproimtion to the volume of the ith ler of wter is i * Î m m nd so its mss is V i r i 4 5 i* m i densit volume r i 4 5 i* 6 i * FIGURE 3 4 The force required to rise this ler must overcome the force of grvit nd so F i m i t i * 568 i * r i Ech prticle in the ler must trvel distnce of pproimtel * i. The work W i done to rise this ler to the top is pproimtel the product of the force nd the distnce * i : W i F i i * 568 i * i * F i - i * To find the totl work done in empting the entire tnk, we dd the contributions of ech of the n lers nd then tke the limit s n l : FIGURE 4 W lim n l n i 568 * i * i 568 d d ( 48 3 ) J Eercises. A 36-lb gorill climbs tree to height of ft. Find the work done if the gorill reches tht height in () seconds (b) 5 seconds. How much work is done when hoist lifts -kg rock to height of 3 m? 3. A vrible force of 5 pounds moves n object long stright line when it is feet from the origin. Clculte the work done in moving the object from ft to ft. 4. When prticle is locted distnce meters from the origin, force of cos 3 newtons cts on it. How much work is done in moving the prticle from to? Interpret our nswer b considering the work done from to.5 nd from.5 to. 5. Shown is the grph of force function (in newtons) tht increses to its mimum vlue nd then remins constnt. How much work is done b the force in moving n object distnce of 8 m? F (N) (m) 6. The tble shows vlues of force function f, where is mesured in meters nd f in newtons. Use the Midpoint Rule to estimte the work done b the force in moving n object from 4 to f ; Grphing clcultor or computer required. Homework Hints vilble t stewrtclculus.com

21 45 CHAPTER 6 APPLICATIONS OF INTEGRATION 7. A force of lb is required to hold spring stretched 4 in. beond its nturl length. How much work is done in stretching it from its nturl length to 6 in. beond its nturl length? 8. A spring hs nturl length of cm. If 5-N force is required to keep it stretched to length of 3 cm, how much work is required to stretch it from cm to 5 cm? 9. Suppose tht J of work is needed to stretch spring from its nturl length of 3 cm to length of 4 cm. () How much work is needed to stretch the spring from 35 cm to 4 cm? (b) How fr beond its nturl length will force of 3 N keep the spring stretched?. If the work required to stretch spring ft beond its nturl length is ft-lb, how much work is needed to stretch it 9 in. beond its nturl length?. A spring hs nturl length cm. Compre the work W done in stretching the spring from cm to 3 cm with the work W done in stretching it from 3 cm to 4 cm. How re W nd relted? W. If 6 J of work is needed to stretch spring from cm to cm nd nother J is needed to stretch it from cm to 4 cm, wht is the nturl length of the spring? 3 Show how to pproimte the required work b Riemnn sum. Then epress the work s n integrl nd evlute it. 3. A hev rope, 5 ft long, weighs.5 lb ft nd hngs over the edge of building ft high. () How much work is done in pulling the rope to the top of the building? (b) How much work is done in pulling hlf the rope to the top of the building? 4. A chin ling on the ground is m long nd its mss is 8 kg. How much work is required to rise one end of the chin to height of 6 m? 5. A cble tht weighs lb ft is used to lift 8 lb of col up mine shft 5 ft deep. Find the work done. 6. A bucket tht weighs 4 lb nd rope of negligible weight re used to drw wter from well tht is 8 ft deep. The bucket is filled with 4 lb of wter nd is pulled up t rte of ft s, but wter leks out of hole in the bucket t rte of. lb s. Find the work done in pulling the bucket to the top of the well. 7. A lek -kg bucket is lifted from the ground to height of m t constnt speed with rope tht weighs.8 kg m. Initill the bucket contins 36 kg of wter, but the wter leks t constnt rte nd finishes drining just s the bucket reches the -m level. How much work is done? 8. A -ft chin weighs 5 lb nd hngs from ceiling. Find the work done in lifting the lower end of the chin to the ceiling so tht it s level with the upper end. 9. An qurium m long, m wide, nd m deep is full of wter. Find the work needed to pump hlf of the wter out of the qurium. (Use the fct tht the densit of wter is kg m 3.). A circulr swimming pool hs dimeter of 4 ft, the sides re 5 ft high, nd the depth of the wter is 4 ft. How much work is required to pump ll of the wter out over the side? (Use the fct tht wter weighs 6.5 lb ft 3.) 4 A tnk is full of wter. Find the work required to pump the wter out of the spout. In Eercises 3 nd 4 use the fct tht 3 wter weighs 6.5 lb ft... 3 m m 3 m 3. 6 ft 4. ; 5. Suppose tht for the tnk in Eercise the pump breks down fter J of work hs been done. Wht is the depth of the wter remining in the tnk? 6. Solve Eercise if the tnk is hlf full of oil tht hs densit of 9 kg m When gs epnds in clinder with rdius r, the pressure t n given time is function of the volume: P P V. The force eerted b the gs on the piston (see the figure) is the product of the pressure nd the re: F r P. Show tht the work done b the gs when the volume epnds from volume to volume is V 8 m frustum of cone V 3 ft 8 ft W V PdV piston hed V 6 ft ft ft m 3 m

22 SECTION 6.5 AVERAGE VALUE OF A FUNCTION In stem engine the pressure P nd volume V of stem stisf the eqution PV.4 k, where k is constnt. (This is true for dibtic epnsion, tht is, epnsion in which there is no het trnsfer between the clinder nd its sur roundings.) Use Eercise 7 to clculte the work done b the engine during ccle when the stem strts t pressure of 6 lb in nd volume 3 3 of in nd epnds to volume of 8 in. 9. () Newton s Lw of Grvittion sttes tht two bodies with msses nd ttrct ech other with force m m F G m m r where r is the distnce between the bodies nd G is the grvittionl constnt. If one of the bodies is fied, find the work needed to move the other from r to r b. (b) Compute the work required to lunch -kg stellite verticll to height of km. You m ssume tht the erth s mss is kg nd is concentrted t its center. Tke the rdius of the erth to be m nd G 6.67 N m kg. 3. The Gret Prmid of King Khufu ws built of limestone in Egpt over -er time period from 58 BC to 56 BC. Its bse is squre with side length 756 ft nd its height when built ws 48 ft. (It ws the tllest mn-mde structure in the world for more thn 38 ers.) The densit of the limestone is bout 5 lb ft 3. () Estimte the totl work done in building the prmid. (b) If ech lborer worked hours d for ers, for 34 ds er, nd did ft-lb h of work in lifting the limestone blocks into plce, bout how mn lborers were needed to construct the prmid? Vldimir Korostshevski / Shutterstock 6.5 Averge Vlue of Function It is es to clculte the verge vlue of finitel mn numbers,,..., : n ve n n T T ve 8 4 t FIGURE But how do we compute the verge temperture during d if infinitel mn temperture redings re possible? Figure shows the grph of temperture function T t, where t is mesured in hours nd T in C, nd guess t the verge temperture, T ve. In generl, let s tr to compute the verge vlue of function f, b. We strt b dividing the intervl, b into n equl subintervls, ech with length b n. Then we choose points *,..., n * in successive subintervls nd clculte the verge of the numbers f *,..., f n * : f * f n * n (For emple, if f represents temperture function nd n 4, this mens tht we tke temperture redings ever hour nd then verge them.) Since b n, we cn write n b nd the verge vlue becomes f * f n * b b f * f n * b n f i * i

23 45 CHAPTER 6 APPLICATIONS OF INTEGRATION If we let n increse, we would be computing the verge vlue of lrge number of closel spced vlues. (For emple, we would be verging temperture redings tken ever minute or even ever second.) The limiting vlue is For positive function, we cn think of this definition s sing re verge height width lim n l b n f * i i b the definition of definite integrl. Therefore we define the verge vlue of f on the intervl, b s f ve b b f d b b f d =ƒ v EXAMPLE Find the verge vlue of the function f on the intervl,. SOLUTION With nd b we hve f ve b b f d d If T t is the temperture t time t, we might wonder if there is specific time when the temperture is the sme s the verge temperture. For the temperture function grphed in Figure, we see tht there re two such times just before noon nd just before midnight. In generl, is there number c t which the vlue of function f is ectl equl to the verge vlue of the function, tht is, f c f ve? The following theorem ss tht this is true for continuous functions. The Men Vlue Theorem for Integrls If f is continuous on, b, then there eists number c in, b such tht f c f ve b b f d f(c)=f ve tht is, b f d f c b c b FIGURE You cn lws chop off the top of (twodimensionl) mountin t certin height nd use it to fill in the vlles so tht the mountin becomes completel flt. The Men Vlue Theorem for Integrls is consequence of the Men Vlue Theorem for derivtives nd the Fundmentl Theorem of Clculus. The proof is outlined in Eercise 5. The geometric interprettion of the Men Vlue Theorem for Integrls is tht, for positive functions f, there is number c such tht the rectngle with bse, b nd height f c hs the sme re s the region under the grph of f from to b. (See Figure nd the more picturesque interprettion in the mrgin note.) v EXAMPLE Since f is continuous on the intervl,, the Men Vlue Theorem for Integrls ss there is number c in, such tht d f c

24 SECTION 6.5 AVERAGE VALUE OF A FUNCTION 453 =+ (, 5) In this prticulr cse we cn find c eplicitl. From Emple we know tht f ve, so the vlue of c stisfies f c f ve (_, ) _ FIGURE 3 f ve = Therefore c So in this cse there hppen to be two numbers c in the intervl, tht work in the Men Vlue Theorem for Integrls. Emples nd re illustrted b Figure 3. v EXAMPLE 3 Show tht the verge velocit of cr over time intervl t, t is the sme s the verge of its velocities during the trip. SOLUTION If s t is the displcement of the cr t time t, then, b definition, the verge velocit of the cr over the intervl is On the other hnd, the verge vlue of the velocit function on the intervl is v ve t t t v t dt t t t t s t dt t t t s t s t s t s t t t so s t s t s t t t c verge velocit (b the Net Chnge Theorem) 6.5 Eercises 8 Find the verge vlue of the function on the given intervl... f 4, f sin 4,, 4, 3. t s 3,, 8 4. t t t,, 3 s3 t 5. f t e sin t cos t,, 6. f sec,, 7. h cos 4 sin,, 8. h u 3 u,, 9. f 3,. f,, 3, 5 ;. f sin sin,, ;. f,, 3. If f is continuous nd f d 8, show tht f tkes on the vlue 4 t lest once on the intervl, Find the numbers b such tht the verge vlue of f 6 3 on the intervl, b is equl to Find the verge vlue of f on, () Find the verge vlue of f on the given intervl. (b) Find c such tht f ve f c. (c) Sketch the grph of f nd rectngle whose re is the sme s the re under the grph of f. 4 6 ; Grphing clcultor or computer required. Homework Hints vilble t stewrtclculus.com

25 454 CHAPTER 6 APPLICATIONS OF INTEGRATION 6. The velocit grph of n ccelerting cr is shown. (km/h) t (seconds) () Use the Midpoint rule to estimte the verge velocit of the cr during the first seconds. (b) At wht time ws the instntneous velocit equl to the verge velocit? 7. In certin cit the temperture (in F) t hours fter 9 AM ws modeled b the function T t 5 4 sin t Find the verge temperture during the period from 9 AM to 9 PM. 8. The velocit v of blood tht flows in blood vessel with rdius R nd length l t distnce r from the centrl is is v r P 4 l R r where P is the pressure difference between the ends of the vessel nd is the viscosit of the blood (see Emple 7 in Section 3.7). Find the verge velocit (with respect to r) over the intervl r R. Compre the verge velocit with the mimum velocit. 9. The liner densit in rod 8 m long is s kg m, where is mesured in meters from one end of the rod. Find the verge densit of the rod.. () A cup of coffee hs temperture 95 C nd tkes 3 minutes to cool to 6 C in room with temperture C. Use Newton s Lw of Cooling (Section 3.8) to show tht the temperture of the coffee fter t minutes is where k.. (b) Wht is the verge temperture of the coffee during the first hlf hour?. In Emple in Section 3.8 we modeled the world popultion in the second hlf of the th centur b the eqution P t 56e.785t. Use this eqution to estimte the verge world popultion during this time period.. If freel flling bod strts from rest, then its displce ment is given b s tt. Let the velocit fter time T be v T. Show tht if we compute the verge of the velocities with respect to t we get v ve v T, but if we compute the verge of the velocities with respect to s we get v ve 3 v T. 3. Use the result of Eercise 83 in Section 5.5 to compute the verge volume of inhled ir in the lungs in one respirtor ccle. 4. Use the digrm to show tht if f is concve upwrd on, b, then f ve f b T t 75e kt +b b 5. Prove the Men Vlue Theorem for Integrls b ppling the Men Vlue Theorem for derivtives (see Section 4.) to the function F f t dt. 6. If f ve, b denotes the verge vlue of f on the intervl, b nd c b, show tht f ve, b c b f ve, c b c b f ve c, b f

26 APPLIED PROJECT CALCULUS AND BASEBALL 455 APPLIED PROJECT CALCULUS AND BASEBALL In this project we eplore three of the mn pplictions of clculus to bsebll. The phsicl interctions of the gme, especill the collision of bll nd bt, re quite comple nd their models re discussed in detil in book b Robert Adir, The Phsics of Bsebll, 3d ed. (New York, ).. It m surprise ou to lern tht the collision of bsebll nd bt lsts onl bout thousndth of second. Here we clculte the verge force on the bt during this collision b first computing the chnge in the bll s momentum. The momentum p of n object is the product of its mss m nd its velocit v, tht is, p mv. Suppose n object, moving long stright line, is cted on b force F F t tht is continuous function of time. () Show tht the chnge in momentum over time intervl t, t is equl to the integrl of F from t to t ; tht is, show tht Btter s bo An overhed view of the position of bsebll bt, shown ever fiftieth of second during tpicl swing. (Adpted from The Phsics of Bsebll) p t p t t F t dt This integrl is clled the impulse of the force over the time intervl. (b) A pitcher throws 9-mi h fstbll to btter, who hits line drive directl bck to the pitcher. The bll is in contct with the bt for. s nd leves the bt with velocit mi h. A bsebll weighs 5 oz nd, in US Customr units, its mss is mesured in slugs: m w t where t 3 ft s. (i) Find the chnge in the bll s momentum. (ii) Find the verge force on the bt. t. In this problem we clculte the work required for pitcher to throw 9-mi h fstbll b first considering kinetic energ. The kinetic energ K of n object of mss m nd velocit v is given b K mv. Suppose n object of mss m, moving in stright line, is cted on b force F F s tht depends on its position s. According to Newton s Second Lw where nd v denote the ccelertion nd velocit of the object. () Show tht the work done in moving the object from position s to position s is equl to the chnge in the object s kinetic energ; tht is, show tht where v v s nd v v s re the velocities of the object t the positions s nd. Hint: B the Chin Rule, s W s F s ds mv mv s m dv dt F s m m dv dt m dv ds (b) How mn foot-pounds of work does it tke to throw bsebll t speed of 9 mi h? ds dt mv dv ds ; Grphing clcultor or computer required

27 456 CHAPTER 6 APPLICATIONS OF INTEGRATION 3. () An outfielder fields bsebll 8 ft w from home plte nd throws it directl to the ctcher with n initil velocit of ft s. Assume tht the velocit v t of the bll fter t seconds stisfies the differentil eqution dv dt v becuse of ir resistnce. How long does it tke for the bll to rech home plte? (Ignore n verticl motion of the bll.) (b) The mnger of the tem wonders whether the bll will rech home plte sooner if it is reled b n infielder. The shortstop cn position himself directl between the outfielder nd home plte, ctch the bll thrown b the outfielder, turn, nd throw the bll to the ctcher with n initil velocit of 5 ft s. The mnger clocks the rel time of the shortstop (ctching, turning, throwing) t hlf second. How fr from home plte should the shortstop position himself to minimize the totl time for the bll to rech home plte? Should the mnger encourge direct throw or reled throw? Wht if the shortstop cn throw t 5 ft s? ; (c) For wht throwing velocit of the shortstop does reled throw tke the sme time s direct throw? APPLIED PROJECT CAS WHERE TO SIT AT THE MOVIES 5 ft ft 9 ft å 4 ft A movie theter hs screen tht is positioned ft off the floor nd is 5 ft high. The first row of sets is plced 9 ft from the screen nd the rows re set 3 ft prt. The floor of the seting re is inclined t n ngle of bove the horizontl nd the distnce up the incline tht ou sit is. The theter hs rows of sets, so 6. Suppose ou decide tht the best plce to sit is in the row where the ngle subtended b the screen t our ees is mimum. Let s lso suppose tht our ees re 4 ft bove the floor, s shown in the figure. (In Eercise 74 in Section 4.7 we looked t simpler version of this problem, where the floor is horizontl, but this project involves more complicted sitution nd requires technolog.). Show tht rccos b 65 b where nd 9 cos 3 sin b 9 cos sin 6. Use grph of s function of to estimte the vlue of tht mimizes. In which row should ou sit? Wht is the viewing ngle in this row? 3. Use our computer lgebr sstem to differentite nd find numericl vlue for the root of the eqution d d. Does this vlue confirm our result in Problem? 4. Use the grph of to estimte the verge vlue of on the intervl 6. Then use our CAS to compute the verge vlue. Compre with the mimum nd minimum vlues of. CAS Computer lgebr sstem required

28 CHAPTER 6 REVIEW Review Concept Check. () Drw two tpicl curves f nd t, where f t for b. Show how to pproimte (b) If S is solid of revolution, how do ou find the crosssectionl res? the re between these curves b Riemnn sum nd 4. sketch the corresponding pproimting rectngles. Then () Wht is the volume of clindricl shell? write n epression for the ect re. (b) Eplin how to use clindricl shells to find the volume (b) Eplin how the sitution chnges if the curves hve of solid of revolution. equtions f nd t, where (c) Wh might ou wnt to use the shell method insted of f t for c d. slicing?. Suppose tht Sue runs fster thn Kth throughout 5-meter rce. Wht is the phsicl mening of the re 5. Suppose tht ou push book cross 6-meter-long tble b eerting force f t ech point from to 6. between their velocit curves for the first minute of the Wht does 6 represent? If f is mesured in newtons, wht re the units for the rce? integrl? 3. () Suppose S is solid with known cross-sectionl res. Eplin how to pproimte the volume of S b Riemnn sum. Then write n epression for the ect volume. 6. () Wht is the verge vlue of function f on n intervl, b? (b) Wht does the Men Vlue Theorem for Integrls s? Wht is its geometric interprettion? 4., bout Eercises 6 Find the re of the region bounded b the given curves. s ;., 4.,,, e 5. Find the volumes of the solids obtined b rotting the region bounded b the curves nd bout the following 3., lines. () The -is 4., 3 (b) The -is (c) 5. sin, 6. Let be the region in the first qudrnt bounded b the curves 3 nd. Clculte the following 6. s,, quntities. () The re of (b) The volume obtined b rotting bout the -is 7 Find the volume of the solid obtined b rotting the region bounded b the given curves bout the specified is. (c) The volume obtined b rotting bout the -is 7. Let be the region bounded b the curves tn, 7., ; bout the -is, nd. Use the Midpoint Rule with n 4 to estimte 8., 3; bout the -is the following quntities. () The re of 9., 9 ; bout (b) The volume obtined b rotting bout the -is., 9 ; bout ; 8. Let be the region bounded b the curves nd 6. Estimte the following quntities.., h (where, h ); bout the -is () The -coordintes of the points of intersection of the curves (b) The re of (c) The volume generted when is rotted bout the -is 4 Set up, but do not evlute, n integrl for the volume of (d) The volume generted when is rotted bout the -is the solid obtined b rotting the region bounded b the given curves bout the specified is.. tn,, 3; bout the -is 9 Ech integrl represents the volume of solid. Describe the solid. 3.,, 4; bout 9. cos d. cos d ; Grphing clcultor or computer required

29 458 CHAPTER 6 APPLICATIONS OF INTEGRATION. sin d d 3. The bse of solid is circulr disk with rdius 3. Find the volume of the solid if prllel cross-sections perpendiculr to the bse re isosceles right tringles with hpotenuse ling long the bse. 9. A tnk full of wter hs the shpe of prboloid of revolution s shown in the figure; tht is, its shpe is obtined b rotting prbol bout verticl is. () If its height is 4 ft nd the rdius t the top is 4 ft, find the work required to pump the wter out of the tnk. ; (b) After 4 ft-lb of work hs been done, wht is the depth of the wter remining in the tnk? 4. The bse of solid is the region bounded b the prbols nd. Find the volume of the solid if the cross-sections perpendiculr to the -is re squres with one side ling long the bse. 4 ft 4 ft 5. The height of monument is m. A horizontl cross-section t distnce meters from the top is n equilterl tringle with side meters. Find the volume of the monument () The bse of solid is squre with vertices locted t,,,,,, nd,. Ech cross-section perpendiculr to the -is is semicircle. Find the volume of the solid. (b) Show tht b cutting the solid of prt (), we cn rerrnge it to form cone. Thus compute its volume more simpl. 7. A force of 3 N is required to mintin spring stretched from its nturl length of cm to length of 5 cm. How much work is done in stretching the spring from cm to cm? 8. A 6-lb elevtor is suspended b -ft cble tht weighs lb ft. How much work is required to rise the elevtor from the bsement to the third floor, distnce of 3 ft? 3. Find the verge vlue of the function f t t sin t on the intervl,. 3. If f is continuous function, wht is the limit s h l of the verge vlue of f on the intervl, h? 3. Let be the region bounded b,, nd b, where b. Let be the region bounded b,, nd b. () Is there vlue of b such tht nd hve the sme re? (b) Is there vlue of b such tht sweeps out the sme volume when rotted bout the -is nd the -is? (c) Is there vlue of b such tht nd sweep out the sme volume when rotted bout the -is? (d) Is there vlue of b such tht nd sweep out the sme volume when rotted bout the -is?

30 Problems Plus. () Find positive continuous function f such tht the re under the grph of f from to t is A t t 3 for ll t. (b) A solid is generted b rotting bout the -is the region under the curve f, where f is positive function nd. The volume generted b the prt of the curve from to b is b for ll b. Find the function f.. There is line through the origin tht divides the region bounded b the prbol nd the -is into two regions with equl re. Wht is the slope of tht line? =8-7 =c FIGURE FOR PROBLEM 3 3. The figure shows horizontl line c intersecting the curve Find the number c such tht the res of the shded regions re equl. 4. A clindricl glss of rdius r nd height L is filled with wter nd then tilted until the wter remining in the glss ectl covers its bse. () Determine w to slice the wter into prllel rectngulr cross-sections nd then set up definite integrl for the volume of the wter in the glss. (b) Determine w to slice the wter into prllel cross-sections tht re trpezoids nd then set up definite integrl for the volume of the wter. (c) Find the volume of wter in the glss b evluting one of the integrls in prt () or prt (b). (d) Find the volume of the wter in the glss from purel geometric considertions. (e) Suppose the glss is tilted until the wter ectl covers hlf the bse. In wht direction cn ou slice the wter into tringulr cross-sections? Rectngulr cross-sections? Cross-sections tht re segments of circles? Find the volume of wter in the glss. L L r r FIGURE FOR PROBLEM 5 r h 5. () Show tht the volume of segment of height h of sphere of rdius r is V 3 h 3r h (See the figure.) (b) Show tht if sphere of rdius is sliced b plne t distnce from the center in such w tht the volume of one segment is twice the volume of the other, then is solution of the eqution where. Use Newton s method to find ccurte to four deciml plces. (c) Using the formul for the volume of segment of sphere, it cn be shown tht the depth to which floting sphere of rdius r sinks in wter is root of the eqution 3 3r 4r 3 s where s is the specific grvit of the sphere. Suppose wooden sphere of rdius.5 m hs specific grvit.75. Clculte, to four-deciml-plce ccurc, the depth to which the sphere will sink. CAS Computer lgebr sstem required 459

31 L FIGURE FOR PROBLEM 6 h =L-h = =_h (d) A hemisphericl bowl hs rdius 5 inches nd wter is running into the bowl t the rte of. in 3 s. (i) How fst is the wter level in the bowl rising t the instnt the wter is 3 inches deep? (ii) At certin instnt, the wter is 4 inches deep. How long will it tke to fill the bowl? 6. Archimedes Principle sttes tht the buont force on n object prtill or full submerged in fluid is equl to the weight of the fluid tht the object displces. Thus, for n object of densit floting prtl submerged in fluid of densit f, the buont force is given b F f t A d, where t is the ccelertion due to grvit nd A is the re of h tpicl cross-section of the object (see the figure). The weight of the object is given b W t L h A d h () Show tht the percentge of the volume of the object bove the surfce of the liquid is f f = C = P B A FIGURE FOR PROBLEM 9 (b) The densit of ice is 97 kg m 3 nd the densit of sewter is 3 kg m 3. Wht percentge of the volume of n iceberg is bove wter? (c) An ice cube flots in glss filled to the brim with wter. Does the wter overflow when the ice melts? (d) A sphere of rdius.4 m nd hving negligible weight is floting in lrge freshwter lke. How much work is required to completel submerge the sphere? The densit of the wter is kg m Wter in n open bowl evportes t rte proportionl to the re of the surfce of the wter. (This mens tht the rte of decrese of the volume is proportionl to the re of the surfce.) Show tht the depth of the wter decreses t constnt rte, regrdless of the shpe of the bowl. 8. A sphere of rdius overlps smller sphere of rdius r in such w tht their intersection is circle of rdius r. (In other words, the intersect in gret circle of the smll sphere.) Find r so tht the volume inside the smll sphere nd outside the lrge sphere is s lrge s possible. 9. The figure shows curve C with the propert tht, for ever point P on the middle curve, the res A nd B re equl. Find n eqution for C.. A pper drinking cup filled with wter hs the shpe of cone with height h nd semiverticl ngle. (See the figure.) A bll is plced crefull in the cup, thereb displcing some of the wter nd mking it overflow. Wht is the rdius of the bll tht cuses the gretest volume of wter to spill out of the cup? 46