4 CHAPTER 6 APPLICATIONS OF INTEGRATION 6. EXERCISES 4 Find the area of the shaded region.. =5-. (4, 4) =. 4. = - = (_, ) = -4 =œ + = + =.,. sin,. cos, sin,, 4. cos, cos, 5., 6., 7.,, 4, 8., 8, 4 4, =_ =e =- 9 Use calculus to find the area of the triangle with the given vertices. 9.,,,,, 6., 5,,, 5, 5 8 Sketch the region enclosed b the given curves. Decide whether to integrate with respect to or. Draw a tpical approimating rectangle and label its height and width. Then find the area of the region. 5. 6. sin, e,, 7., 8., 4 9...,.,., 4. cos, cos, 5. tan, sin, 6., 9,,,,, s, 4 6 Evaluate the integral and interpret it as the area of a region. Sketch the region... sin cos d 4 s d 4 Use the Midpoint Rule with n 4 to approimate the area of the region bounded b the given curves.. sin 4, cos 4, 4. s 6,, ; 5 8 Use a graph to find approimate -coordinates of the points of intersection of the given curves. Then find (approimatel) the area of the region bounded b the curves. 5. sin, 4 7. s,, 8. 8,,, 9., 4. 4, 9 6. e, 7., 8. cos, 4
SECTION 6. AREAS BETWEEN CURVES 4 9. Use a computer algebra sstem to find the eact area enclosed b the curves 5 6 4 and. 4. Sketch the region in the -plane defined b the inequalities, and find its area. (c) Which car is ahead after two minutes? Eplain. (d) Estimate the time at which the cars are again side b side. 4. Racing cars driven b Chris and Kell are side b side at the start of a race. The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race. Use the Midpoint Rule to estimate how much farther Kell travels than Chris does during the first ten seconds. A B t (min) 4. The widths (in meters) of a kidne-shaped swimming pool were measured at -meter intervals as indicated in the figure. Use the Midpoint Rule to estimate the area of the pool. 4. A cross-section of an airplane wing is shown. Measurements of the height of the wing, in centimeters, at -centimeter intervals are 5.8,., 6.7, 9., 7.6, 7.,.8,.5, 5., 8.7, and.8. Use the Midpoint Rule to estimate the area of the wing s cross-section. 44. If the birth rate of a population is b t e.4t people per ear and the death rate is d t 46e.8t people per ear, find the area between these curves for t. What does this area represent? 45. t v C v K t v C v K 6 69 8 7 75 86 7 8 8 9 46 5 9 86 98 4 54 6 9 5 6 7 6. 5. 7. 6.8 5.6 4.8 4.8 cm Two cars, A and B, start side b side and accelerate from rest. The figure shows the graphs of their velocit functions. (a) Which car is ahead after one minute? Eplain. (b) What is the meaning of the area of the shaded region? 46. The figure shows graphs of the marginal revenue function R and the marginal cost function C for a manufacturer. [Recall from Section 4.7 that R and C represent the revenue and cost when units are manufactured. Assume that R and C are measured in thousands of dollars.] What is the meaning of the area of the shaded region? Use the Midpoint Rule to estimate the value of this quantit. Rª() Cª() 5 ; 47. The curve with equation is called Tschirnhausen s cubic. If ou graph this curve ou will see that part of the curve forms a loop. Find the area enclosed b the loop. 48. Find the area of the region bounded b the parabola, the tangent line to this parabola at,, and the -ais. 49. Find the number b such that the line b divides the region bounded b the curves and 4 into two regions with equal area. 5. (a) Find the number a such that the line a bisects the area under the curve, 4. (b) Find the number b such that the line b bisects the area in part (a). 5. Find the values of c such that the area of the region bounded b the parabolas c and c is 576. 5. Suppose that c. For what value of c is the area of the region enclosed b the curves cos, cos c, and equal to the area of the region enclosed b the curves cos c,, and? 5. For what values of m do the line m and the curve enclose a region? Find the area of the region.
4 CHAPTER 6 APPLICATIONS OF INTEGRATION we would have obtained the integral V h EXAMPLE 9 A wedge is cut out of a circular clinder of radius 4 b two planes. One plane is perpendicular to the ais of the clinder. The other intersects the first at an angle of along a diameter of the clinder. Find the volume of the wedge. SOLUTION If we place the -ais along the diameter where the planes meet, then the base of the solid is a semicircle with equation s6, 4 4. A crosssection perpendicular to the -ais at a distance from the origin is a triangle ABC, as shown in Figure 7, whose base is and whose height is. Thus the cross-sectional area is BC tan s6 s s6 L h h d L h C A s6 s s6 6 s A 4 B =œ 6- C and the volume is V 4 A d 4 4 4 6 d s s 4 6 d s 6 4 A B 8 s FIGURE 7 For another method see Eercise 64. M 6. Volumes EXERCISES 8 Find the volume of the solid obtained b rotating the region bounded b the given curves about the specified line. Sketch the region, the solid, and a tpical disk or washer..,,, ; about the -ais., ; about the -ais.,,, ; about the -ais 4. s5,,, 4; about the -ais. 4,, ; about the -ais., s ; about. e,, ; about. sec, ; about 4.,,, ; about 5. s,, 9; about the -ais 6. ln,,, ; about the -ais 7.,, ; about the -ais 8. 4, 5 ; about the -ais 9., ; about the -ais 5., ; about 6., s ; about 7., ; about 8.,,, 4; about
SECTION 6. VOLUMES 4 9 Refer to the figure and find the volume generated b rotating the given region about the specified line. 9. about OA. about OC. about AB. about BC. about OA 4. about OC 5. about AB 6. about BC 7. about OA 8. about OC 9. about AB. about BC 6 Set up, but do not evaluate, an integral for the volume of the solid obtained b rotating the region bounded b the given curves about the specified line.. tan,, ; about. 4, 8 6; about., sin, ; about 4., sin, ; about 5., ; about 6. cos, cos, ; about 4 ; 7 8 Use a graph to find approimate -coordinates of the points of intersection of the given curves. Then use our calculator to find (approimatel) the volume of the solid obtained b rotating about the -ais the region bounded b these curves. 7. cos, 8. sin, C(, ) 9 4 Use a computer algebra sstem to find the eact volume of the solid obtained b rotating the region bounded b the given curves about the specified line. 9. sin,, ; O 4., e ; T =œ T 4 e e about T = A(, ) B(, ) about 4. 4 8 d 44. 45. A CAT scan produces equall spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained onl b surger. Suppose that a CAT scan of a human liver shows cross-sections spaced.5 cm apart. The liver is 5 cm long and the cross-sectional areas, in square centimeters, are, 8, 58, 79, 94, 6, 7, 8, 6, 9, and. Use the Midpoint Rule to estimate the volume of the liver. 46. A log m long is cut at -meter intervals and its crosssectional areas A (at a distance from the end of the log) are listed in the table. Use the Midpoint Rule with n 5 to estimate the volume of the log. 47. (a) If the region shown in the figure is rotated about the -ais to form a solid, use the Midpoint Rule with n 4 to estimate the volume of the solid. 4 4 6 8 (b) Estimate the volume if the region is rotated about the -ais. Again use the Midpoint Rule with n 4. 48. (a) A model for the shape of a bird s egg is obtained b rotating about the -ais the region under the graph of f a b c d s Use a to find the volume of such an egg. (b) For a Red-throated Loon, a.6, b.4, c., and d.54. Graph f and find the volume of an egg of this species. 49 6 Find the volume of the described solid S. 49. A right circular cone with height h and base radius r 5. A frustum of a right circular cone with height h, lower base radius R, and top radius r r cos d m (m) A ( m ) (m) A ( ).68 6.5.65 7.55.64 8.5.6 9.5 4.58.48 5.59 4 44 Each integral represents the volume of a solid. Describe the solid. 4. cos d 4. 5 d R h
5 CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION N Figure 8 shows the interpretation of the arc length function in Eample 4. Figure 9 shows the graph of this arc length function. Wh is s negative when is less than? s() P = - 8 ln s()= + 8 ln - FIGURE 8 FIGURE 9 8. EXERCISES. Use the arc length formula () to find the length of the curve 5,. Check our answer b noting that the curve is a line segment and calculating its length b the distance formula.. Use the arc length formula to find the length of the curve s,. Check our answer b noting that the curve is part of a circle. 6 Set up, but do not evaluate, an integral for the length of the curve.. cos, 4. e, 5. ln, 6. s sin (s ) 7. e, e 8. ln e, a b, a ; 9 Find the length of the arc of the curve from point P to point Q. 9., P(, ), Q(, ) 5., 4. 4, P, 5, Q 8, 8 6. a b 7 8 Find the length of the curve. 7. 6, 8. 4 4,, ; Graph the curve and visuall estimate its length. Then find its eact length..,., 6 9. 5, 6. 4, 8 4. s,. ln cos,. ln sec, 9 4 4. cosh, 6 Use Simpson s Rule with n to estimate the arc length of the curve. Compare our answer with the value of the integral produced b our calculator.. e, 4. s, 5. sec, 5 6. ln,
SECTION 8. ARC LENGTH 5 ; 7. (a) Graph the curve s 4, 4. (b) Compute the lengths of inscribed polgons with n,, and 4 sides. (Divide the interval into equal subintervals.) Illustrate b sketching these polgons (as in Figure 6). (c) Set up an integral for the length of the curve. (d) Use our calculator to find the length of the curve to four decimal places. Compare with the approimations in part (b). ; 8. Repeat Eercise 7 for the curve sin 9. Use either a computer algebra sstem or a table of integrals to find the eact length of the arc of the curve ln that lies between the points, and, ln.. Use either a computer algebra sstem or a table of integrals to find the eact length of the arc of the curve 4 that lies between the points, and,. If our has trouble evaluating the integral, make a substitution that changes the integral into one that the can evaluate. the distance traveled b the pre from the time it is dropped until the time it hits the ground. Epress our answer correct to the nearest tenth of a meter. 8. The Gatewa Arch in St. Louis (see the photo on page 56) was constructed using the equation 9..49.96 cosh.9765 for the central curve of the arch, where and are measured in meters and 9.. Set up an integral for the length of the arch and use our calculator to estimate the length correct to the nearest meter. A manufacturer of corrugated metal roofing wants to produce panels that are 8 in. wide and in. thick b processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verif that the sine curve has equation sin 7 and find the width w of a flat metal sheet that is needed to make a 8-inch panel. (Use our calculator to evaluate the integral correct to four significant digits.). Sketch the curve with equation and use smmetr to find its length.. (a) Sketch the curve. (b) Use Formulas and 4 to set up two integrals for the arc length from, to,. Observe that one of these is an improper integral and evaluate both of them. (c) Find the length of the arc of this curve from, to 8, 4.. Find the arc length function for the curve with starting point P,. ; 4. (a) Graph the curve 4,. (b) Find the arc length function for this curve with starting point P (, 7 ). (c) Graph the arc length function. w 4. (a) The figure shows a telephone wire hanging between two poles at b and b. It takes the shape of a catenar with equation c a cosh a. Find the length of the wire. ; (b) Suppose two telephone poles are 5 ft apart and the length of the wire between the poles is 5 ft. If the lowest point of the wire must be ft above the ground, how high up on each pole should the wire be attached? 8 in in 5. Find the arc length function for the curve sin s with starting point,. 6. A stead wind blows a kite due west. The kite s height above ground from horizontal position to 8 ft is given b 5 4 5. Find the distance traveled b the kite. 7. A hawk fling at 5 m s at an altitude of 8 m accidentall drops its pre. The parabolic trajector of the falling pre is described b the equation 8 until it hits the ground, where is its height above the ground and is the horizontal distance traveled in meters. Calculate 45 _b 4. Find the length of the curve st dt b 4 ; 4. The curves with equations n n, n 4, 6, 8,...,are called fat circles. Graph the curves with n, 4, 6, 8, and to see wh. Set up an integral for the length L k of the fat circle with n k. Without attempting to evaluate this integral, state the value of lim k l L k.
APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES A9 Eercises. (a) 8 (b) 5.7. =ƒ =ƒ 6 6. 5. 7. f is c, f is b, 4 f t dt is a 9 9. 7.. 76 5. 4 7. Does not eist 9. sin.. 5. 7. ( sin t ln C s 4 C C 9. e s C. ln cos C. 4 ln 4 C 5. ln sec C 7. 64 9. s sin C 4. 5 4. F 45. t 4 cos 8 47. (e e s ) 49. 4 s d 4s 55..898 57. Number of barrels of oil consumed from Jan.,, through Jan., 8 59. 7,4 6. 6. c.6 65. f e e 7. 5. 7. 6 4 =9 (6, 9) = =œ (, ) = = PROBLEMS PLUS N PAGE 4.. f 5. 7. e 9.,. (a) n n (b) b b b a a a 7. (s ) 9. 64 5 CHAPTER 6 = (4, ) EXERCISES 6. N PAGE 4.. e e 5. 9.5 7. 9. ln 6 59.. 7 5. ln 7. 9. 8.. 5. 7. ln 9. 6.5. s..647 5.,.9;.4 7. 8.8 9. s6 9 4. 7 ft 4. 4 cm 45. (a) Car A (b) The distance b which A is ahead of B after minute (c) Car A (d) t. min 47. 4 5 s 49. 4 5. 6 5. m ; m ln m EXERCISES 6.. 9 N PAGE 4 = =- = =. 6. ( 4 s) π _, = =œ π, = =+sec = (, ) = = =
A9 APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES 5. 6 55. = = = (, ) e =e _ (, _) 7. 9 = (, ) = _ =_ 7. 6 =4(-)@ 7 = -4+7 (, 4) (, 4) 9. 9... 5. 7. 9.. tan d. 5. sin d s [5 (s ) ] d 7..88,.884;.78 9. 8 4. Solid obtained b rotating the region cos, about the -ais 4. Solid obtained b rotating the region above the -ais bounded b and 4 about the -ais 45. cm 47. (a) 96 (b) 88 49. r h 5. 5. b h 55. cm 57. 4 59. 6. 6. (a) 8 R r sr d (b) 65. (b) r h 67. 69. 8 r sr sr d EXERCISES 6. N PAGE 46. Circumference, height ; 5. 7 5 4 s h (r h) 8 5 5 r 4 r R 7 5. 768 7. 6 5. 7 5 7. 8 9. 5 4. ln d. sin 4 d 5. 4 ssin d 7..68 9. Solid obtained b rotating the region 4, about the -ais. Solid obtained b rotating the region bounded b (i),, and, or (ii),, and about the line.. 5. 7. 9. 4 ln 4 4 4. 4 4. r 45. r h EXERCISES 6.4 N PAGE 44 8. 588 J. 9 ft-lb 5. 8 J 7. 4 ft-lb 5 9. (a) 4.4 J (b).8 cm. W W 875. (a) 65 ft-lb (b) 4 ft-lb 5. 65, ft-lb 7. 857 J 9. 45 J..6 6 J..4 5.. m 9. Gm m a b 5 ft-lb 5
APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES A97 57. sin s4 sin ln(sin s4 sin ) C 9. s ln( s ). 6. No. 7 (45s45 s) 5. a 6. (a).95444 (b).995 (c).947 7. 9.754 9..5796 65. (a).48, n 68 (b).674, n 6. 67. 8.6 mi. 69. (a).8 (b).7867,.646 (c) n 6 [ln(s ) s] 56 7. C 7. 75. 7. (a) a (b) 45 sa PROBLEMS PLUS PAGE 5 9. (a) b a b sin (sa b a) N sa b. About.85 inches from the center. 7. f. b b a a b a e (b) a ab sin (sb a b). sin sb ( s5) a. b c f s f d. a CHAPTER 8 6 4 [4 ln(s7 4) 4 ln(s ) s7 4s] 4 r EXERCISES 8. N PAGE 5. 4s5. s 5. 4 s9 4 6 d sin d 6 7. 4 (8s8 ) 9. 4.. ln(s ) 5. ln 7. s e s ln(s e ) ln(s ) 46 9. s ln( s).. 5.584 5..56969 7. (a), (b) L 4, L 6.4, L 4 7.5 EXERCISES 8. N PAGE 547. (a) 87.5 lb ft (b) 875 lb (c) 56.5 lb. 6 lb 5. 6.7 4 N 7. 9.8 N 9.. 4 lb. ah. 5.7 5 N 5. (a) 4 N (b) 5 N 7. (a) 5.6 lb (b) 5.6 4 lb (c) 4.88 4 lb (d). 5 lb 9..5 5 N. ;. ; ; (, 7 ) 5.,.6 7. 9. (, 9 ) 9 e, e 4 s 4.. (, ) 4(s ), 4(s ) 5. 6; 6; ( 8, ) 7..78,. 4. (, ) 45. r h (c) 4 s 4 4 (d) 7.7988 d 9.. 6 s5 ln( ( s5)) s ln( s) EXERCISES 8.4 N PAGE 55. $8,. $4,866,9. 5. $47.5 7. $, 9. 77; $7,75 k b k a k. (6s 8) $9.75 million. k b k a k 5..9 4 cm s 7. 6.6 L min 9. 5.77 L min. s 7 [ 9 s] 5. s (s ) 7. 9. m 9. 9.6 in. 4..4 EXERCISES 8. N PAGE 57. (a) 4 s 6 6 d (b) s 6 6 d. (a) tan d (b) d 5. 7 (45s45 ) 7. 98 EXERCISES 8.5 N PAGE 56. (a) The probabilit that a randoml chosen tire will have a lifetime between, and 4, miles (b) The probabilit that a randoml chosen tire will have a lifetime of at least 5, miles. (a) f for all and f d (b) 8 s.5 5. (a) (b) 7. (a) for all and f d (b) 5. (a) e 4.5. (b) e.5.55 (c) If ou aren t served within minutes, ou get a free hamburger.. 44% 5. (a).668 (b) 5.% 7..9545