(2)(5 x) dx = 10x x x 3 dx = 1 4 x (x+1)( x1/2 ) dx = = 1 2 { } 1 2 { } = m3 4.

Transcription

1 Odd Answers: Chapter Five Contemporary Calculus PROBLE ANSWERS Chapter Five Section.. volume (.. ) + (.. ) + (.. ) 7. v v (π...) + (π.. ) + (π.. ) 9π. volume (9..) + (..) + (..). 7. (ase)(height)(thickness) ()( x) dx x x ( 9) () in 8. (ase)(height)(thickness) (x)( x ) dx x dx x 8 in. in 9. (ase)(height)(thickness) (x+)( x/ ) dx x / + x / dx { x/ + x/ } { + } { + } 8 m. π(radius) (thickness) π( ( x ) ) dx π 8x + x dx π (x x + x ) π { } π { } π m. Solids A and B have equal volumes. Why?. (ase)(height)(thickness) ( x )( x ) dx x dx ( x ) 9 cuic units.. (side)(side)(thickness) ( x )( x ) dx x dx x ( ) ( ) cuic units.. π(radius) (thickness) π( x ) dx π ( x ( ) x/ + x ) 8 π π(radius) (thickness) π( x ) dx π (9 x + x) dx π ( 9x ( ) x/ + x ) π( + 8) π() π 7.7 cuic units. π π π 8. π(radius) (thickness) π( sin(x) ) dx π ( sin(x) + sin (x) ) dx π{ x + cos(x) + x sin(x) } π π { π + ( ) + π } π{ + + } π π. cuic units.

2 Odd Answers: Chapter Five Contemporary Calculus 9. (ase)(height)(thickness) ( x )( x ) dx x dx ( x ) 8 () 8 () cuic units. π/.. cos(x) dx cuic units.. π(radius) (thickness) π( x ) dx π( ) dx π x dx π( x x) π( ) π( ) π. cuic units.. π 8.8 cuic units.. π. π 7. Rr π 9. on your own. (a) H L () H L (c) ratio. (a) BL () Volume BL (c) ratio Section ( ) ( ) 7.97 feet ( ) + ( 7) + ( ) + ( 9).8 feet (/) + + (/) + + (/).. (a) (,) to (,): + () y ' : L + ( ) dx dx x (same as in part (a) ) 7. (a) (,) to (, ): + () x ', y ' : L + ( ) dt dt t 9. y ' x x / : L + ( y ' ) dx + ( x ) dx + x dx. y ' x ( + x) / () / () /.787 x : + (y ') + (x + x ) x + + x ( x + x )

3 Odd Answers: Chapter Five Contemporary Calculus L + ( y ' ) dx x + x dx x + x dx x x ( () () ) ( (). y ' x x : + (y ') + (x 8 + x 8 ) x x 8 ( x + x ) L + ( y ' ) dx x + x dx x x \ () ).. y ' x: L ( () + ( y ' ) dx () ) ( () + ( x ) dx () ).88 + x dx.79 (using calculator). y ' x : L + ( y ' ) dx + 9x dx.8 (using calculator) 7. y ' x / : L ( y ' ) dx + x dx 8.8 (using calculator) e 8. y ' x : L e + ( y ' ) dx + x dx. (using calculator) π/ 9. y ' cos(x): (a) L π/ () L π/ + cos (x) dx.8 (using calculator) + cos (x) dx.8 (using calculator). x '. sin(t), y '. cos(t) π L (. sin(t) ) + (. cos(t) ) dt π 9. sin (t) +. cos (t) dt. (using calculator). x '. sin(t), y '. cos(t) π L. sin (t) +. cos (t) dt.8 (using calculator). x ' t. sin(t) + cos(t), y ' t. cos(t) + sin(t). Then π ( x ' ) + ( y ' ) t + (check it) so L + t dt. (using calculator)

4 Odd Answers: Chapter Five Contemporary Calculus. same x ' and y ' as in #. L + t dt. (using calculator). x ' R. ( cos(t)) and y ' R. sin(t): ( x ' ) + ( y ' ) R. ( cos(t) ) π L π R. ( cos(t)) dt R cos(t) dt 8R (Actually, 8R ). mile,8 feet at π feet per revolution so mile 8 ft π ft/rev 8.8 revolutions. From #, total distance ( 8.8 rev.)( 8 feet/rev) 7.7 feet (.7 miles).. (Why?) 7. y ' x: total length + x dx.88 (using calculator) T (a) Find T so + x dx (.88) 8.9. By experimenting on a calculator, T.77. A () Find A so + x dx (.88). and B B so + x dx (.88).. By experimenting on a calculator, A. and B.. 8., 9. on your own.. (a) A: SA x π()() π B: SA x π()() 8π () A: SA y π()() π B: SA y π()() π. (a) A: SA x π()() 8π B: SA x π()( + ) π () A: SA y π()() π B: SA y π(.)( ) π. (a) aout y : A: SA π()() π B: SA π()( ) π () aout x : A: SA π()() π B: SA π(8.)( ) 8π. The largest surface area occurs when the midpoint is farthest from the line of rotation, the x axis, when θ 9 o. Then SA π(dist. to x axis)(length) π()() π. 7. y ' x : SA y πx + ( y ' ) dx πx + ( x ) dx πx + x dx.7 (using calculator)

5 Odd Answers: Chapter Five Contemporary Calculus 9. y ' x: SA x. y ' x : SA x. y ' x: SA y πy + ( y ' ) dx πy + (y ') dx πx + (y ') dx π( x ) + x dx. (using calculator) π( x ) + 9x dx. (using calculator) π x + x dx.77 (using calculator) or use a u sustitution with u + x : SA y. on your own Section. u7 u π u / 8 du.77.. Fig.. Weight (volume)(density) (.. y)(. ), distance raised 7 y i, so Work (7 y)( 7 y i ). 7 Total work (7 y)(7) dy 7( 7y ) 7 y 8,7 ft ls.. Slice at height y: weight (.. y)() y l, distance raised y ft. (a) Total work ( y)() dy ( y ) y,8 ft ls. () The top cuic feet of water corresponds to dimensions of ft. long, ft wide, and Weight of slice y, distance raised y (where y goes from. ft. to ft.) Total work ( y)() dy ( y ) y,888 ft ls... (c) With HP pump: it takes 8 ()() minutes. min 8.7 seconds to empty the tank. With / HP pump: 8.7 / 7. seconds. Both pumps do the same amount of work ut in different times and thus have different power.. Slice at height y: weight (π. r. y)() (π.. y)() π y l, distance raised y ft. (a) Total work ( y)(π) dy π( y ) y π(8),9 ft ls. () weight of slice π y l, distance raised ( + ) y 8 y ft. Total work (8 y)(π) dy π( 8y ) y π 7. ft ls. (c) First find the work needed to empty the top feet; using integration of part (a) with new limits.. ft. high.

6 Odd Answers: Chapter Five Contemporary Calculus work ( y)(π) dy π( y y ) π(.) 88 ft ls. Knowing that HP pump works at a rate of, ft ls/minute, then it takes 88.7 minutes. seconds. 7. Fig.. Use similar triangles: slice at height y has weight (volume)(density) (7.. y. y)( 8 ) 8y y and distance raised y. Total work ( y)(8y) dy 8( y ) y, ft ls. 8. Fig.. Each slice (perpendicular to the y axis) is a disk with volume π(radius) (thickness) π ( y ) y so weight (volume)(density) ( π ( y ) y )( ) π y y. (a) Distance to top 8 y so 8 Total work (8 y)( π y ) dy π 8 8y y dy π( 8 y y ) 8 π. 8,. π,7 ft ls. () The limits of integration are now to 8: 8 work (8 y)( π y ) dy π 8 8y y dy π( 8 y y ) π,8 ft ls. H 9. Find H so π 8y y dy { total work done in 8(a) } {,. π },7.π. H π 8y y dy π{ 8 y ) H y π H { 8 H }. Some "exploring" with a calculator shows that if H.9 then π H { 8 H },7.π, the value we want. One person should remove the top feet of grain and the other should remove the ottom.9 feet.

7 Odd Answers: Chapter Five Contemporary Calculus 7. Fig.. Slice at height y: weight (volume)(density) ( π r y)(.) ( π ( y ) y)(.) π y y. (a) Distance raised y so work ( y)( π y ) dy π { y y } π ( 8 ). ft ls. () The only change is the distance raised: distance y. work ( y)( π y ) dy π { y y } π ( 8 ),879.8 ft ls.. Fig.. Slice at height y: weight (volume)(density) ( π r y)(.) ( π ( y ) y)(.). π ( y ) y. distance y. work ( y)(.π )( y ) dy.π y y + y dy.π{ y 8y y + } y.π( 89. ) 7, ft ls.. { work for (a) } > { work for () } > { work for (c) }. For container (c), most of the water is raised only a small distance, ut for container (a), most of the water is raised a larger distance. 7. Work area under the curve of force vs. position. (a) work ()() + ()() + ()() + ()() 8 ft ls. () work ()() + ()() + ()() ft ls. 9. F kx so x kx and k. (a) work x dx x 7 in oz. () work x dx x 8 in oz.. Work { area under graph of force vs. length }. Use the graph in Fig. to approximate the area. (a) From x to x, area ( cm) { avg. height of aout g } g cm.

8 Odd Answers: Chapter Five Contemporary Calculus 8 () From x 8 to x, area ( 8 cm) { g } + ( ) { g) 97. g cm.. F kx so k. and k /. + x dx 9 x.8 in ls. work R+h R P. Work f(x) dx a R x dx 9. { +h } +h () () x dx 9. { x +h } (a) h : work 9. { } 7, mile pounds. () h : work 9. { }, mile pounds. (c) h : work 9. { }, mile pounds. 7. Assume you weigh P pounds. R+h Work f(x) dx R P a R x dx +h () P x dx.. 7 P { x } +h +h }.. 7 P { (a) h : work.. 7 P { } mile pounds. () h : work.. 7 P { } mile pounds. (c) h : work.. 7 P { + } mile pounds. What happens when h is really large? 9. f(x) kx and we know that f. when x so. work a f(x) dx a (.)x dx { x } { a a }. k and k.. (a) from x to x : work { }. ft l. () from x to x : work { } 9 ft l. (c) from x to x.: work {. } 9 ft l. t. Work ta f(t) ( dx/dt ) + ( dy/dt ) dt π π t sin (t) + cos (t) dt t dt t π π.

9 Odd Answers: Chapter Five Contemporary Calculus 9 t. Work ta f(t) ( dx/dt ) + ( dy/dt ) dt t t + dt. (Use the sustitution u t + to find an antiderivative, and then evaluate.). "Unroll" the region to get a triangle with ase π(radius) π() π and height f(π) π. The "area" of the triangle is (ase)(height) (π)(π) π. (This is the same as prolem.) Section.. (a) + +. o ()() + ()() + ()() 8. x o 8. () new x o ()() + ()() + ()() + (8)(? ) (? ) 8 so? 8.. (c) x o 8 + (? )() +? so 7 + (? ) 8 + (? ) and (? ) so?.. (a) + +. x y ()() + ()() + ()() 8. x y () new x new y ()() + ()() + ()(). 8 + (? ) + + (?) + so? 8.. so?.8.. See Fig. 9. A: mass ()()(density). Center of mass (., ) B: mass ()()(density). Center of mass (,.) Total mass +. y ()(.) + ()(). x ()() + ()(.) 7. y x.7 and y x 7.. Notice that the center of mass is not "in" the region. 7. See Fig.. A: mass ()(). Center of mass (.,.) B: mass ()(). Center of mass (,.) C: mass ()(). Center of mass (.,.) Total mass ++. y ()(.)+()()+()(.). x ()(.)+()(.)+()(.) 7. y x.8. y x See Fig.. A: mass (8)(). Center of mass (, )

10 Odd Answers: Chapter Five Contemporary Calculus B: mass.π( ).8. Center of mass (, 8+.() ) (, 8.8) C: mass.π( ).. Center of mass (+.(), ) (.7, ) Total mass.. y ()()+(.8)()+(.)(.7) 9.8. x ()() + (.8)(8.8) + (.)() y x y x Fig.. x x. f(x) dx f(x) dx x. x dx x dx x x 9. y ( f(x) ) dx f(x) dx x dx x dx x x... Fig.. Because of symmetry aout the y axis, x. y x a x. { f (x) g (x) } dx { ( x ) } dx { x x } { ( ) ( + )}. a { x } dx { x x } { (8 8 ) ( 8+8 ) }.7. y x..7.

11 Odd Answers: Chapter Five Contemporary Calculus. Fig.. Because of symmetry aout the y axis, x. y x. x a { f (x) } dx ( x ) dx { x 8 x + x } { ( + ) ( + )} 7.7 a { x } dx { x x } { (8 8 ) ( 8+8 ) }.7. x y Fig.. f(x) g(x) dx 9 x dx x x. y " x. { f(x) g(x) } dx x. (9 x ) dx x x x { f (x) g (x) } dx { (9 x) () } dx { 8 8x + x 9 } dx a { 7x 9x + x } { 8+9) 7. y x 8... y x Fig.. a 9 f(x) dx x dx x/ y 9 x. { f(x) } dx x. ( x ) dx 9 x/ x a 9 f (x) dx ( x ) dx x y x y x Fig.. a f(x) dx e e x dx e. x e x (e e) ( )

12 Odd Answers: Chapter Five Contemporary Calculus y x. { f(x) } dx x. ( e e x ) dx e. x e x (x ) ( e ) (+).9 x a { f (x) g (x) } dx e e x dx { e. x ex } {e e } { }.97. y x.9.9. y x (a) h o ()() + ( x )( x )( ) + ( x )( ) + x + x. () Find minimum of h in part (a): calculate d h d t, set d h d t and solve for x to find critical points. d h x( + x) ( + d t x ) ( + x). If d h d t then x( + x) ( + x so x ± + 9. inches.. Empty glass: weight oz., y 8 cm. Full glass (liquid only): weight oz., y cm, volume ( cm)( cm)(8 cm) 8 cm. density of liquid weight volume oz. 8 cm oz. 8 cm Partially full glass (liquid only): x height of liquid, y x cm, weight ( cm)( cm)(x cm)( 8 oz. cm ) x oz h height of cm of glass containing x cm of liquid 7. and 9. On your own. ( oz)( cm) + (x oz)( x cm) oz + x oz + x + x cm.. Center of gravity is feet aove the ground and is raised to a height of feet: distance c.g. raised 8 feet. work (force)(distance) ()(8), ft l.. C.g. is raised + feet. Work ()(), ft l.

13 Odd Answers: Chapter Five Contemporary Calculus. (a) Volume aout x axis A. π. y. π. π ft. Surface area aout x axis P. π. y 8. π. π ft. () Volume aout y axis A. π. x. π. π ft. Surface area aout y axis P. π. x 8. π. 8π ft. (c) Volume aout line A. π. (dist. of c.g. from line). π. π ft. Surface area aout line P. π. (dist of c.g. from line) 8. π. π ft. 7. (a) Volume aout x axis A. π. y (π ). π. π ft. Surface area aout x axis P. π. y π(). π. π ft. () Volume aout y axis A. π. x π. π. π ft. Surface area aout y axis P. π. x π. π. π ft. (c) Volume aout line A. π. (dist. of c.g. from line) π. π. π ft. Surface area aout line P. π. (dist of c.g. from line) π. π. π ft. 9. Each rectangle has area 8 ft, perimeter ft., and centroid ft. from the line of rotation. Volume of each π(radius)(area) π( ft)(8 ft ) 8π ft.8 ft. Surface area of each π(radius)(perimeter) π( ft)( ft) 7π ft. ft. Section. Liquid Pressure. A: d. x. () dx d x dx d x d. Fig.. B: d. x. (x ) dx d x x dx. Fig.. C: d ( x x ) d. d. x. ( x) dx. d D: d. x. () dx d.

14 Odd Answers: Chapter Five Contemporary Calculus. Rectangular end: d. x. () dx d x dx d x d. Triangular end (Fig. ): d. x. ( )( x) dx d x x dx d { x x ) d. The total force is unchanged if the length is douled.. Fig.. d. ( y). ( y ) dy d y / y / dy d { ( y/ ) y/ } d { ( (8) ) () } d.. All three have the same total force against their sides. 7. The one with the largest perimeter: not (a), proaly () 9. Fig.. (a) 8 () Kinetic Energy d. x. ( π) dx πd x d. x. ( π) dx πd x 8 πd (),πd. πd (),7πd.. (a) g. v rev/sec ( π(cm) )/sec 9π cm/sec. KE v ( g) ( 9π cm/s ) 8, π ergs 799,8 ergs. () g. v rev/sec ( π(cm) )/sec π cm/sec. KE v ( g) ( π cm/s ), π ergs,, ergs.. Fig.. KE i ( x )(. πx ) π x x π x dx 8π x KE.7. π ergs ergs.. Fig. 7. ( g)/( cm). g/cm. KE i (. x )( πx ).π x x KE.π x dx.π x Total KE {,7 π ergs } 8,98 ergs.,7 π ergs,9 ergs.

15 Odd Answers: Chapter Five Contemporary Calculus 7. (a) m i π r x π x x, v i rev/sec ( π radians )/sec πx radians/sec. KE i m i ( v i ) ( π x x )( πx ) π x x KE π x dx π ( ) x 9 π ( ) 7 π () m i π x x, v i rev/sec ( π radians )/sec πx radians/sec. KE i m i ( v i ) ( π x x )( πx ) 7 π x x KE 7 π x dx 7 π ( ) x 8 π ( ), π 9. density g/cm. (a) Fig. 8. m i (area)( g/cm ) ( x)( g/cm ) x. v i rev/sec ( π x radians )/sec π x radians/sec. KE i m i ( v i ) ( x )( πx ) π x x KE π x dx 8 π x 8 π (7), π Total KE {, π ), π () Fig. 9. m i (area)( g/cm ) ( x)( g/cm ) 8 x. v i π x radians/sec. KE i m i ( v i ) ( 8 x )( πx ) π x x KE π x dx 8 π x 8 π (), π Total KE {, π ), π Volumes using tues: volume a π(radius)(height)(thickness). Fig.. a π(radius)(height)(thickness) π( x )( x ) dx (put u x, then du x dx) u π u / $ # " du ' & ) π ( u % ( ) u/ u u $ "# ' $ & )() " "# ' & )() # % ( % (.

16 Odd Answers: Chapter Five Contemporary Calculus. Fig.. a π(radius)(height)(thickness) π π( - x )( x x ) dx + " π( - x )( x x) dx "x + x " 8x dx + π x " x + 8xdx π{ } + π{ 7 }.. Fig.. a π(radius)(height)(thickness) π( - x )( x ) dx π x " 8 + x dx π{ ln(x) " 8 x + x } π{ - + ln() } Fig.. a π(radius)(height)(thickness) π( x )( x ln(x) ) dx 8.8 (Using calculator.) 9. Fig.. π(radius)(height)(thickness) a / π( x )( 9 x x) dx finish on your own. Voting. a votes for cand. A, votes for cand. A, c votes for cand. C.. a votes for cand. B, votes for cand. C, c votes for cand. C.. Fig. 9: B wins.. Fig. : A wins.. (a) Fig. a: A wins. () Fig. : C wins. (c) Fig. c: B wins. Notice that in candidate elections, A loses to B and A loses to C. In the candidate election, A wins.. (a) B wins. () B wins (c) A wins 7. (a) A wins. () A wins (c) looks like C wins 8. (a) B wins. () A wins (c) looks like A wins 9. on your own