EXERCISES 6.1. Cross-Sectional Areas. 6.1 Volumes by Slicing and Rotation About an Axis 405
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1 6. Volumes b Slicing and Rotation About an Ais 5 EXERCISES 6. Cross-Sectional Areas In Eercises and, find a formula for te area A() of te crosssections of te solid perpendicular to te -ais.. Te solid lies between planes perpendicular to te -ais at = - and =. In eac case, te cross-sections perpendicular to te -ais between tese planes run from te semicircle = - - to te semicircle = -. a. Te cross-sections are circular disks wit diameters in te -plane. d. Te cross-sections are equilateral triangles wit bases in te -plane.. Te solid lies between planes perpendicular to te -ais at = and =. Te cross-sections perpendicular to te -ais between tese planes run from te parabola = - to te parabola =. a. Te cross-sections are circular disks wit diameters in te -plane. b. Te cross-sections are squares wit bases in te -plane. c. Te cross-sections are squares wit diagonals in te -plane. (Te lengt of a square s diagonal is times te lengt of its sides.) b. Te cross-sections are squares wit bases in te -plane. c. Te cross-sections are squares wit diagonals in te -plane. d. Te cross-sections are equilateral triangles wit bases in te -plane.
2 6 Capter 6: Applications of Definite Integrals Volumes b Slicing Find te volumes of te solids in Eercises Te solid lies between planes perpendicular to te -ais at = and =. Te cross-sections perpendicular to te ais on te interval are squares wose diagonals run from te parabola = - to te parabola =.. Te solid lies between planes perpendicular to te -ais at = - and =. Te cross-sections perpendicular to te -ais are circular disks wose diameters run from te parabola = to te parabola = Te solid lies between planes perpendicular to te -ais at = - and =. Te cross-sections perpendicular to te -ais between tese planes are squares wose bases run from te semicircle = - - to te semicircle = Te solid lies between planes perpendicular to te -ais at = - and =. Te cross-sections perpendicular to te -ais between tese planes are squares wose diagonals run from te semicircle = - - to te semicircle = Te base of a solid is te region between te curve = sin and te interval [, p] on te -ais. Te cross-sections perpendicular to te -ais are a. equilateral triangles wit bases running from te -ais to te curve as sown in te figure. sin b. squares wit bases running from te -ais to te curve. 8. Te solid lies between planes perpendicular to te -ais at = -p>3 and = p>3. Te cross-sections perpendicular to te -ais are a. circular disks wit diameters running from te curve = tan to te curve = sec. b. squares wose bases run from te curve = tan to te curve = sec. 9. Te solid lies between planes perpendicular to te -ais at = and =. Te cross-sections perpendicular to te -ais are circular disks wit diameters running from te -ais to te parabola = 5.. Te base of te solid is te disk +. Te cross-sections b planes perpendicular to te -ais between = - and = are isosceles rigt triangles wit one leg in te disk.. A twisted solid A square of side lengt s lies in a plane perpendicular to a line L. One verte of te square lies on L. As tis square moves a distance along L, te square turns one revolution about L to generate a corkscrew-like column wit square cross-sections. a. Find te volume of te column. b. Wat will te volume be if te square turns twice instead of once? Give reasons for our answer.. Cavalieri s Principle A solid lies between planes perpendicular to te -ais at = and =. Te cross-sections b planes perpendicular to te -ais are circular disks wose diameters run from te line = > to te line = as sown in te accompaning figure. Eplain w te solid as te same volume as a rigt circular cone wit base radius 3 and eigt. Volumes b te Disk Metod In Eercises 3 6, find te volume of te solid generated b revolving te saded region about te given ais. 3. About te -ais. About te -ais 3 3
3 6. Volumes b Slicing and Rotation About an Ais 7 5. About te -ais 6. About te -ais Find te volumes of te solids generated b revolving te regions bounded b te lines and curves in Eercises 7 about te -ais. 7. =, =, = 8. = 3, =, = 9. = 9 -, =. = -, =.. = cos, p>, =, = = sec, =, = -p>, = p> In Eercises 3 and, find te volume of te solid generated b revolving te region about te given line. 3. Te region in te first quadrant bounded above b te line =, below b te curve = sec tan, and on te left b te -ais, about te line =. Te region in te first quadrant bounded above b te line =, below b te curve = sin, p>, and on te left b te -ais, about te line = Find te volumes of te solids generated b revolving te regions bounded b te lines and curves in Eercises 5 3 about te -ais. 5. Te region enclosed b = 5, =, = -, = 6. Te region enclosed b = 3>, =, = 7. Te region enclosed b = sin, p>, = 8. Te region enclosed b = cos sp>d, -, = 9. = >s + d, =, =, = 3 3. = >s + d, =, = Volumes b te Waser Metod Find te volumes of te solids generated b revolving te saded regions in Eercises 3 and 3 about te indicated aes. 3. Te -ais 3. Te -ais cos tan sin cos tan Find te volumes of te solids generated b revolving te regions bounded b te lines and curves in Eercises about te -ais. 33. =, =, = 3. =, =, = 35. = +, = = -, = = sec, =, -p> p> 38. = sec, = tan, =, = In Eercises 39, find te volume of te solid generated b revolving eac region about te -ais. 39. Te region enclosed b te triangle wit vertices (, ), (, ), and (, ). Te region enclosed b te triangle wit vertices (, ), (, ), and (, ). Te region in te first quadrant bounded above b te parabola =, below b te -ais, and on te rigt b te line =. Te region in te first quadrant bounded on te left b te circle + = 3, on te rigt b te line = 3, and above b te line = 3 In Eercises 3 and, find te volume of te solid generated b revolving eac region about te given ais. 3. Te region in te first quadrant bounded above b te curve =, below b te -ais, and on te rigt b te line =, about te line = -. Te region in te second quadrant bounded above b te curve = - 3, below b te -ais, and on te left b te line = -, about te line = - Volumes of Solids of Revolution 5. Find te volume of te solid generated b revolving te region bounded b = and te lines = and = about a. te -ais. b. te -ais. c. te line =. d. te line =. 6. Find te volume of te solid generated b revolving te triangular region bounded b te lines =, =, and = about a. te line =. b. te line =. 7. Find te volume of te solid generated b revolving te region bounded b te parabola = and te line = about a. te line =. b. te line =. c. te line = B integration, find te volume of te solid generated b revolving te triangular region wit vertices (, ), (b, ), (, ) about a. te -ais. b. te -ais. Teor and Applications 9. Te volume of a torus Te disk + a is revolved about te line = b sb 7 ad to generate a solid saped like a dougnut
4 8 Capter 6: Applications of Definite Integrals and called a torus. Find its volume. (Hint: -a a - d = pa >, since it is te area of a semicircle of radius a.) 5. Volume of a bowl A bowl as a sape tat can be generated b revolving te grap of = > between = and = 5 about te -ais. a. Find te volume of te bowl. b. Related rates If we fill te bowl wit water at a constant rate of 3 cubic units per second, ow fast will te water level in te bowl be rising wen te water is units deep? 5. Volume of a bowl a. A emisperical bowl of radius a contains water to a dept. Find te volume of water in te bowl. b. Related rates Water runs into a sunken concrete emisperical bowl of radius 5 m at te rate of. m 3 >sec. How fast is te water level in te bowl rising wen te water is m deep? 5. Eplain ow ou could estimate te volume of a solid of revolution b measuring te sadow cast on a table parallel to its ais of revolution b a ligt sining directl above it. 53. Volume of a emispere Derive te formula V = s>3dpr 3 for te volume of a emispere of radius R b comparing its cross-sections wit te cross-sections of a solid rigt circular clinder of radius R and eigt R from wic a solid rigt circular cone of base radius R and eigt R as been removed as suggested b te accompaning figure. R R R a 56. Designing a plumb bob Having been asked to design a brass plumb bob tat will weig in te neigborood of 9 g, ou decide to sape it like te solid of revolution sown ere. Find te plumb bob s volume. If ou specif a brass tat weigs 8.5 g>cm 3, ow muc will te plumb bob weig (to te nearest gram)? 57. Ma-min Te arc = sin, p, is revolved about te line = c, c, to generate te solid in Figure 6.6. a. Find te value of c tat minimizes te volume of te solid. Wat is te minimum volume? b. Wat value of c in [, ] maimizes te volume of te solid? T c. Grap te solid s volume as a function of c, first for c and ten on a larger domain. Wat appens to te volume of te solid as c moves awa from [, ]? Does tis make sense psicall? Give reasons for our answers. (cm) 6 7 (cm) (cm) 9 cm deep 6 (cm) sin 5. Volume of a cone Use calculus to find te volume of a rigt circular cone of eigt and base radius r. c 55. Designing a wok You are designing a wok fring pan tat will be saped like a sperical bowl wit andles. A bit of eperimentation at ome persuades ou tat ou can get one tat olds about 3 L if ou make it 9 cm deep and give te spere a radius of 6 cm. To be sure, ou picture te wok as a solid of revolution, as sown ere, and calculate its volume wit an integral. To te nearest cubic centimeter, wat volume do ou reall get? s L = cm 3.d FIGURE 6.6 c
5 6. Volumes b Slicing and Rotation About an Ais An auiliar fuel tank You are designing an auiliar fuel tank tat will fit under a elicopter s fuselage to etend its range. After some eperimentation at our drawing board, ou decide to sape te tank like te surface generated b revolving te curve = - s >6d, -, about te -ais (dimensions in feet). a. How man cubic feet of fuel will te tank old (to te nearest cubic foot)? b. A cubic foot olds 7.8 gal. If te elicopter gets mi to te gallon, ow man additional miles will te elicopter be able to fl once te tank is installed (to te nearest mile)?
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