Calculus 152 Take Home Test 2 (50 points)

Transcription

1 Calculus 5 Take Home Test (5 points) Due Tuesday th November. The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I epect you to comply with the following conditions.. I will not talk to any person about any part of this test either directly or indirectly.. I will not use the internet or any mathematical based program such as Scientific Notebook.. I can use my class notes; solutions manual; tet book; homework assignments and graphing calculator. In order to make this test a fair reflection of my ability I promise to comply with the above conditions. Students Signature: Print Name: Page

2 . Use the Shell Method find the volume of the solid formed by rotating the curve y about the y-ais from to.. Use the Shell Method to find the volume of the solid formed by rotating the region bounded by the curves y and y about the line y Page

3 .(a) Find the arc length of y 5 6 on [,].(b) Find the arc length of y (e + e ) on [ln ( ), ln ()] Page

4 .(a) Find the area of the surface generated when the curve y is rotated about the -ais from to..(a) Find the area of the surface generated when the curve y from to. + 8 is rotated about the -ais Page

5 5. Suppose a force of 6 N is required to stretch and hold a spring 5 cm from its equilibrium position. (a) Assume the spring obeys Hooke s law, find the Spring Constant k. (b) How much work is required to compress the spring cm from its equilibrium position? 6. Page 5

6 7. A small dam is drawn below. Assume that the water level is at the top of the dam, find the total force on the face of the dam. 8. Evaluate the following integrals using the method of Integration By Parts. Page 6

7 (a) e d (b) ln( ) d 9. Evaluate the integral e sin( ) d (Hint the I method) using the method of Integration By Parts. Page 7

8 . Evaluate the integralsin. cos d. Evaluate the Integral sin d Page 8

9 . Evaluate the Integral tan d. Show that the integral d. will become the integral tan sec. d When you use the substitution tan. Do not evaluate the integral only show the process. Page 9

10 .(a) Find the integral 6 d using sec θ.(b) Find the integral d by using the Trig substitution sin θ Page

11 5.(a) Evaluate the integral 6 d using partial fractions. 5.(b) Evaluate the integral d using partial fractions. ( )( ) Page

12 6. For each of the following integrals state the integration method you would use, if it is u-substitution or -substitution then indicate the substitution used, if it is integration by parts then indicate the function you would use for f() and g(). Integration Method u-sub or -sub used or state f(), g() if Integration of Parts (a) e d (b) 9d (c) ln. d (d) sin e cos d (e) tan d e (f) e d (g) 9d (h) e sin d Page

13 Solutions:. Use the Shell Method find the volume of the solid formed by rotating the curve y about the y-ais from to. Volume b a f ( ) d ( ) d ( ) d { } ( () () ) () Volume 8. Use the Shell method of cylindrical shells to find the volume of the solid formed by rotating the region bounded by the curves and y about the line y y y d V c yf ( y) dy ( y)( y y ) dy ( y)( y y ) dy (y y y y ) dy typical radius y y y 5 y y y y y 5 y 5 () () () () () 5 V () 5 Page

14 .(a) Find the arc length of y 5 6 on [,] Length b + (f ()) d a + (5) d 6 d ( 6 ) 6 () 6 () Length 6.(b) Find the arc length of y (e + e ) on [ln ( ), ln ()] y y dy d (e + e ) e + e e e ( dy d ) ( e e ) + ( dy d ) + ( e e ) ( e e ) + e + e e + + e Length Length b + (f ()) d a ln () e + + e d ln () ln () ( e + e ) d ln () ln () ln () ( e + e ) d n() ( e e ln ( )) ( eln () eln( ) ) ( e ln () eln( ) ) eln () + eln () eln( )) eln( ) e ln () e ln ( ) Page

15 .(a) Find the area of the surface generated when the curve y is rotated about the -ais from to. y y dy d ( dy d ) ( ) ( dy d ) ( dy d ) + ( dy d ) + + b S f() + (f ()) d a + d 8 + d 8 + d 8 ( + ) d 8 ( ( + ) ) 8( ( + ) ( + ) ) 8( 8 5 ) 8( ) S 6 (6 5 5) Page 5

16 .(b) Find the area of the surface generated when the curve y from to. + 8 is rotated about the -ais y dy d ( dy d ) ( ) ( dy d ) ( + ) b S f() + (f ()) d a ( 8 + ) ( + ) d ( 8 + ) ( + ) d ( ) d ( ) d ( ) (( ) ( )) (( + 5 ) ( 8 + )) (( ) ( 8 + )) (( ) 9 8 ) ( ) ( 79 5 ) S Page 6

17 5. Suppose a force of 6 N is required to stretch and hold a spring 5 cm from its equilibrium position. (a) Assume the spring obeys Hooke s law, find the Spring Constant k. F() k 6 k(.5) k (b) How much work is required to compress the spring cm from its equilibrium position? b Work k d a. d (. ) (.) Work 8 NM 6. In this question you can assume ρ, and g 9.8 (a) W W 8 ρga(y)w(y)dy 8 8 (,)(9.8)()ydy 9,ydy [96, y ] 5,, y 8 y (b) It is not true since ρga(y)w(y)dy ρga(y)w(y)dy 8 Page 7

18 7. A small dam is drawn below. Assume that the water level is at the top of the dam, find the total force on the face of the dam. y W(y) the width of a typical strip at depth y is So W(y) W(y) y Using Pythagoras to find in terms of y Total Force b 98 W(y)D(y)dy a 98 y y dy 98 y y dy Use u-sub u y 98 y u 98 u du y du 98 u du 98( u ) ( 9,6 ( y ) ) ( 9,6 ( y ) y y ) 9,6 9,6 [( ) ( ) ] [8] Total Force 56,8, or Total Force 5. X 7 N Page 8

19 8. Evaluate the following integrals using the method of Integration By Parts. (a) e d f()g() f ()G()d e e d e d (b) ln() d ln() d e e d e (f()g() f ()G()d) e ( e ( e )d) e e + e d e e + e + C f()g() f ()G()d ln () d ln () d ln() 6 + C 9. Evaluate the integral e sin( ) d using the method of Integration By Parts. I e sin() d f()g() f ()G()d Where f() e and g() sin () e cos() e cos()d e cos() + e cos()d e cos() + (f()g() f ()G()d) Where f() e and g() cos () e cos() + (e sin() e sin() d) e cos() + e sin() e sin() d) I e cos() + e sin() I 5 I e cos() + e sin() I 5 ( e cos() + e sin()) + C 5 e cos() + 5 e sin() + C Page 9

20 . Evaluate the integral sin cos d sin cos d sin cos cos d let u sin sin cos ( sin ) d u ( u ) cos u ( u )du (u u )du u 5 u5 + C sin 5 sin5 + C du cos ( sin 5 sin5 ) ( sin 5 sin5 ) ( sin 5 sin5 ) du d du cos cos du cos d sin d sin cos d. Evaluate the Integral sin d sin d (sin )(sin ) d ( cos ) ( cos ) d ( cos )( cos ) d ( cos + cos ) d ( cos + ( + cos )) d sin d ( cos + + cos ) d ( cos + cos ) d ( sin sin ) ( sin + sin ) ( ) 8 8 Page

21 . Evaluate the Integral tan d tan d tan tan d tan ( sec ) d (tan sec tan ) d tan sec d tan d u du v dv u ln v tan ln (cos ) ( tan ln (cos ) ) ( tan ln (cos )) ( tan ln (cos )) ( () ln )) ( () ln( )) ( (ln ln )) ( ) tan d ln + ln + ln ln + ln Page

22 . Show that the integral + d will become the integral tan θ sec θ dθ When you use the substitution tan. Do not evaluate the integral only show the process. converting d. into tan sec. d d. d. tan sec. d tan tan 8 tan tan sec t. d d dθ sec d sec d 8 tan (tan ) sec t. d 8 tan sec sec. d 8tan sec sec. d tan sec. d + d tan θ sec θ dθ Page

23 .(a) Find the integral 6 d using sec θ sec θ sec θ d sec θ tan θ sec θ dθ d sec θ tan θ dθ HYP ADJ d sec θ tan θ dθ 6 ( sec θ) ( sec θ) 6 θ HYP ADJ OPP 6 sec θ tan θ dθ 6sec θ 6sec θ 6 8sec θ tan θ dθ 6sec θ 6tan θ 8sec θ tan θ 6sec θ( tan θ) dθ dθ 6 sec θ cos θ dθ 8 8 sin θ + C d 6 + C 6 8 Page

24 (b). Find the integral d by using the Trig substitution sin θ sin θ sin θ d dθ d cos θ cos θdθ sin θ OPP HYP d (sin θ) (sin θ) cosθ d θ 8sin θ sin θ cosθ dθ 8sin θ cos θ cosθ dθ cos θ ( + cosθ) 8sin θ cosθ cosθ dθ sin θ ( cosθ) 6sin θ cos θ dθ cos θ ( + cosθ) 6 ( cosθ) ( + cosθ) dθ ( cos θ )dθ ( cosθ)dθ ( cosθ)dθ ( cosθ)dθ θ sinθ + C d sin ( ) + ( ) + C θ HYP ADJ OPP Since θ sin ( ) sinθ sinθcosθ sinθ sinθcosθ(cos θ ) sinθ ( ( ) ( ( ) ) ) ( ) θ sinθ + C sin ( ) + ( ) + C 5.(a) Evaluate the integral 6 (+)( ) d using partial fractions. Use the cover up method to get 6 (+)( ) 6 d (+)( ) 6 d (+)( ) A + B (+) ( ) + (+) ( ) ( + )d (+) ( ) ln( + ) + ln( ) + C Page

25 5.(b) Evaluate the integral ( +)(+) d using partial fractions. ( +)(+) A+B ( +) + C (+) (A+B) (+) + ( +) (+) C ( +) (+) ( +) ( +)(+) (A+B)(+)+C( +) ( +)(+) (A + B)( + ) + C( + ) Choose (A + B)( + ) + C( + ) ( ) (A( ) + B)( + ) + C(( ) + ) (A( ) + B)()+ C( + ) 5 5C C So (A + B)( + ) ( + ) Choose () (A() + B)( + ) ( + ) (B)() B So (A + )( + ) ( + ) Choose () (A() + )( + ) ( + ) (A + )() ( + ) A + 5 A A A d ( +)(+) A+B ( +) + C (+) d + ( +) (+) d ( +) + ( +) (+) d ln u + tan ln( + ) + C d ( +)(+) ln( + ) + tan ln( + ) + C Page 5

26 6. For each of the following integrals state the integration method you would use, if it is u-substitution or -substitution then indicate the substitution used, if it is integration by parts then indicate the function you would use for f() and g(). Integration Method u-sub or -sub used or state f(), g() if Integration of Parts (a) e d Integration By Parts f() g() e (b) 9d -substitution sinθ (c) ln. d Integration By Parts f() ln g() cos (d) sin e d U-substitution u cos (e) tan d Integration By Parts f() tan - g() e (f) e d U-substitution u + e (g) 9d U-substitution u 9 (h) e sin d Integration By Parts f() e g() sin Page 6