MAC 2311 Chapter 5 Review Materials Topics Include Areas Between Curves, Volumes (disks, washers, and shells), Work, and Average Value of a Function

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1 MAC Chapter Review Materials Topics Include Areas Between Curves, Volumes (disks, washers, and shells), Work, and Average Value of a Function MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the area of the shaded region. ) f() = g() = ) 0 (, 8) 0 (0, 0) (-, -) -0 A) B) 8 C) 78 D) 97 ) = ) = A) 9 B) C) D)

2 ) = + - = - (, ) ) A) 9 B) 9 C) 8 D) ) = - ) A) = - B) C) 8 D)

3 ) ) = cos -π -π π π = -cos - - A) B) + π C) - π D) + π Find the area enclosed b the given curves. ) = -, = - A) B) C) D) 7 ) 7) =, = A) B) C) D) 7) 8) = - sin, = sin, 0 π 8) A) B) C)8 D) 9) = csc, = cot, = π, and = π 9) A) π B) π C)π D) π 0) Find the area of the region in the first quadrant bounded on the left b the line = π and on the 0) right b the curves = tan and = cot. (Round to four decimal places.) A).09 B) 0. C) 0.88 D) 0.09 Find the volume of the described solid. ) The solid lies between planes perpendicular to the -ais at = 0 and = 9. The cross sections perpendicular to the -ais between these planes are squares whose bases run from the parabola = - to the parabola =. A) B) 0 C) 79 D) 8 )

4 ) The base of the solid is the disk +. The cross sections b planes perpendicular to the -ais between = - and = are isosceles right triangles with one leg in the disk. A) 8 B) C) 80 D) ) Find the volume of the solid generated b revolving the shaded region about the given ais. ) About the -ais ) = - + A) 8 π C) 8π D) 8π ) About the -ais ) = sec π π A) π C)8π D)π ) About the -ais ) = A) 7 π C)π D) 8π

5 ) About the -ais ) π = tan π π A) π - 0 0π C)0π + π D) 0π - 0π 7) About the -ais 7) = 9 - A) 7 8 π C)9π D) π Find the volume of the solid generated b revolving the region bounded b the given lines and curves about the -ais. 8) =, = 0, =, = 8) A) 7 π C) 7 π D) π 9) =, = 0, = 0, = A) π 9) 0) =, = 0, =, = 0) A) πln C) π D) π ) = 8 -, = 0, = 0, = 9 A) 97 8π C) 8π D) π )

6 ) = 7csc, = 0, = π, = π ) A) 9 98π C) 7π D) π ) =, =, = 0 A)π C)8π D)π ) ) =, =, = 0 A) π B) π C)0π D) π ) ) = +, = + A) π 0 ) ) = sin, =, = 0 to = π 8 ) A) π 8 - π B) π 8 + π C) π 8 - π D) π 8-7) = 9cos (π), = 9, = -0., = 0. A) 7π C)8π D) 8 π 7) Find the volume of the solid generated b revolving the region about the given line. 8) The region bounded above b the line = 9, below b the curve = 9 -, and on the right b the line =, about the line = 9 A) π 8) 9) The region in the first quadrant bounded above b the line =, below b the curve =, and on the left b the -ais, about the line = A) π 9) 0) The region bounded above b the line =, below b the curve = cos (π), on the left b the line = -0., and on the right b the line = 0., about the line = A) π C)π - D) π - 8 0) ) The region in the first quadrant bounded above b the line + =, below b the -ais, and on the left b the -ais, about the line = - A) 0 8π C) 9π D) π )

7 ) The region in the first quadrant bounded above b the line =, below b -ais, and on the right b the line =, about the line = - A) π ) Find the volume of the solid generated b revolving the region about the -ais. ) The region enclosed b =, = 0, = -, = A) 8 π C) 8 π D) π ) ) The region enclosed b =, = 0, =, = ) A) 7 7 π ) The region enclosed b = sin, 0 π, = 0 ) A) π C)π D) π ) The region enclosed b the triangle with vertices (0, 0), (, 0), (, ) A) π ) 7) The region in the first quadrant bounded on the left b =, on the right b the line =, and below b the -ais A) π 7) 7

8 Use the shell method to find the volume of the solid generated b revolving the shaded region about the indicated ais. 8) About the -ais 8) = / A) 0 π 9) About the -ais 9) = = = - A) 8 π 8

9 0) About the -ais 0) = = - /9 A) 8 7π ) About the -ais ) 7 = - A) π ) About the -ais ) =sin( ).8 A) 9π C)π D) π 9

10 Use the shell method to find the volume of the solid generated b revolving the region bounded b the given curves and lines about the -ais. ) = 8, = - 8, = ) A) 8 π ) =, = A) 0 π C) π D) π ) ) = -, =, = 0 A) 8 π ) ) =, = 0, =, = A) π C)π D) π ) 7) =, = 0, =, = 7) A) 8π C)π D) π Use the shell method to find the volume of the solid generated b revolving the region bounded b the given curves and lines about the -ais. 8) =, = -, = A) π C) π D) π 8) 9) = 9 -, = 0 A) π 9) 0) =, =, = A)π C)8π D) π 0) ) =, = A) π B) π 7 C) π D) π ) ) =, = A) 8 π C) π D) π ) 0

11 Use the shell method to find the volume of the solid generated b revolving the region bounded b the given curves about the given lines. ) =, = 0, = ; revolve about the -ais ) A) π C) D) π ) = 9 -, = 9, = ; revolve about the line = 9 A) π ) Use the shell method to find the volume of the solid generated b revolving the shaded region about the indicated line. ) About the line = ) = A) π ) About the line = - ) = - A) π

12 7) About the line = - 7) = (solid) = (dashed) A) 0 0 π 8) About the line = - 8) = + = 9 - A) π 9) About the line = 9) = (solid) = / (dashed) (0, ) A) π

13 Find the volume of the solid generated b revolving the region about the given ais. Use the shell or washer method. 0) The triangle with vertices (0, 0), (0, ), and (, ) about the line = 0) A) 0 π C) 8 π D) π ) The region bounded b = 7, = 7, and = 0 about the line = 7 A) π ) ) The region bounded b = 7, = 7, and = 0 about the -ais A) 7 π C) 7 π D) 7 0 π ) ) The region bounded b = - and = about the line = A) 8 π C)π D) π ) ) The region in the first quadrant bounded b = - and the -ais about the -ais A) 8 π C) π D)π ) Solve the problem. ) A bead is formed from a sphere of radius b drilling through a diameter of the sphere with a drill bit of radius. Find the volume of the bead. A)8π C) 8 π D) π ) ) An auiliar fuel tank for a helicopter is shaped like the surface generated b revolving the curve = -,-, about the -ais (dimensions are in feet). How man cubic feet of fuel will 9 the tank hold to the nearest cubic foot? A) B) C) D) 0 7) The spring of a spring balance is 7.0 in. long when there is no weight on the balance, and it is 7. in. long with.0 lb hung from the balance. How much work is done in stretching it from 7.0 in. to a length of. in.? A). in.-lb B) in.-lb C) 70 in.-lb D) 0 in.-lb ) 7) 8) It took 90 J of work to stretch a spring from its natural length of m to a length of 7 m. Find the spring's force constant. A) 80 N/m B) 0 N/m C)80 N/m D) 90 N/m 8) 9) It takes a force of,000 lbto compress a spring from its free height of in. to its full compressed height of 7 in. How much work does it take to compress the spring the first inch? A) 00 in.-lb B) 00 in.-lb C) 0,000 in.-lb D) 00 in.-lb 9)

14 70) A force of N will stretch a rubber band cm. Assuming Hooke's law applies, how much work is done on the rubber band b a N force? A) 0. J B) 9000 J C)0. J D) 0.9 J 70) 7) Find the work done in winding up a 0-ft cable that weighs.00 lb/ft. A) 00 ft lb B),00 ft lb C),000 ft lb D),000 ft lb 7) 7) The gravitational force (in lb) of attraction between two objects is given b F = k/, where is the distance between the objects. If the objects are ft apart, find the work required to separate them until the are 7 ft apart. Epress the result in terms of k. A) 7 k B) 0 k C) k D) k 7) 7) A rescue cable attached to a helicopter weighs lb/ft. A 00-lb man grabs the end of the rope and is pulled from the ocean into the helicopter. How much work is done in lifting the man if the helicopter is 0 ft above the water? A) 900 ft lb B) 7800 ft lb C)00 ft lb D) 00 ft lb 7) 7) A construction crane lifts a bucket of sand originall weighing 0 lb at a constant rate. Sand is lost from the bucket at a constant rate of 0. lb/ft. How much work is done in lifting the sand 0 ft? (Neglect the weight of the bucket.) A) 0 ft lb B) 700 ft lb C)87 ft lb D) 8 ft lb 7) 7) A vertical right circular clindrical tank measures 0 ft high and 0 ft in diameter. It is full of oil weighing 0 lb/ft. How much work does it take to pump the oil to the level of the top of the tank? Give our answer to the nearest ft lb. A),88,9 ft lb B) 9,78 ft lb C),708 ft lb D),79,9 ft lb 7) A vertical right circular clindrical tank measures 8 ft high and 8 ft in diameter. It is full of oil weighing 0 lb/ft. How long will it take a (/)-horsepower (hp) motor (work output 7 ft lb/sec) to pump the oil to the level of the top of the tank? Give our answer to the nearest minute. A) 7 min B) min C)99 min D) min 77) A swimming pool has a rectangular base ft long and ft wide. The sides are ft high, and the pool is half full of water. How much work will it take to empt the pool b pumping the water out over the top of the pool? Assume that the water weighs. lb/ft. Give our answer to the nearest ft lb. A) 8,0 ft lb B),0 ft lb C),0 ft lb D) 8,80 ft lb 78) A conical tank is resting on its ape. The height of the tank is 8 ft, and the radius of its top is ft. The tank is full of gasoline weighing lb/ft. How much work will it take to pump the gasoline to the top? Give our answer to the nearest ft lb. A),7 ft lb B),0 ft lb C)9,09 ft lb D) 0 ft lb 7) 7) 77) 78)

15 79) A tank is designed b revolving the parabola =, 0, about the -ais. The tank, with dimensions in meters, is filled with water weighing 9800 N/m. How much work will it take to empt the tank b pumping the water to the tank's top? Give our answer to the nearest J. A),,0 J B),80 J C),90,8 J D),00 J 79) Find the fluid force eerted against the verticall submerged flat surface depicted in the diagram. Assume arbitrar units, and call the weight-densit of the fluid w. 80) 80) A) 0 w B) w C)w D) w 8) 8) A)w B)w C)w D) 8w Solve the problem. 8) One end of a pool is a vertical wall ft wide. What is the force eerted on this wall b the water if it is ft deep? The densit of water is. lb/ft. A) 90 lb B) 0 lb C)0 lb D),000 lb 8) A right triangular plate of base 8 m and height m is submerged verticall, as shown below. Find the force on one side of the plate. (w = 9800 N/m ) 8) 8) m m 8 m A) 00,000 N B) 0,000 N C)0,000 N D) 0,000 N 8) Find the force on one side of a cubical container cm on an edge if the container is filled with mercur. The densit of mercur is kn/m. A) 8. N B) 800 N C)0 N D).7 N 8)

16 8) A semicircular plate ft in diameter sticks straight down into fresh water with the diameter along the surface. Find the force eerted b the water on one side of the plate. A) 9. lb B) 78.8 lb C).8 lb D) 99. lb 8) 8) A rectangular swimming pool has a parabolic drain plate at the bottom of the pool. The drain plate is shaped like the region between = and the line = from = - to =. The pool is 0 ft 8) b 0 ft and 8 ft deep. If the pool is being filled at a rate of 00 ft /hr, what is the force on the drain plate after hours of filling? Round our answer to two decimal places if necessar. A). lb B) 9. lb C)08 lb D) 97.7 lb

17 Answer Ke Testname: CHAPTER AREAS BETWEEN CURVES, VOLUMES, WORK, AND AVERAGE VALUE OF A FUNCTION ) D ) A ) A ) D ) D ) A 7) B 8) C 9) D 0) D ) D ) B ) B ) B ) B ) D 7) B 8) B 9) A 0) D ) C ) B ) A ) A ) B ) A 7) D 8) C 9) A 0) C ) D ) A ) D ) A ) D ) B 7) D 8) C 9) D 0) C ) B ) D ) C ) C ) A ) C 7) C 8) D 9) D 7

18 Answer Ke Testname: CHAPTER AREAS BETWEEN CURVES, VOLUMES, WORK, AND AVERAGE VALUE OF A FUNCTION 0) B ) B ) A ) B ) C ) B ) D 7) A 8) C 9) C 0) D ) C ) B ) D ) B ) C ) D 7) C 8) D 9) D 70) D 7) D 7) A 7) A 7) C 7) B 7) A 77) D 78) B 79) A 80) B 8) B 8) A 8) B 8) A 8) C 8) B 8