( )dt F. ( ) = y 2 sin y. ( ) = t 2 sint dt. ( ) = 1+ 2x. ( ) = 1+ 2t dt. ( ) = cos t 2. ( ) = cos x 2 ( ) ( ) = arctan 1 x 1 x 2 = 1 x 2 arctan 1 x

Transcription

1 Section. The Fndamental Theorem of Calcls Part Soltions. g y g y y ( ) t sint dt ( ) y sin y. g g ( ) + t dt ( ) +. F ( )dt ( ) cos t F ( )dt F ( ) cos t ( ) ( ) cos. h ( ) arctant dt ( ) arctan arctan h 5. y cost dt y cos t cos

2 Section. Definite Integrals and The Sbstittion Rle Soltions ( ) 5 d. + + d 6 d. sec ( t )dt t d dt 6 6 ( + ) 5 d sec t ( )dt 6 5 d sec d tan ( 6 78 ) 8 9 tan tan. e d d d. sinθ cos θ dθ cosθ d sinθdθ e d sinθ cos θ dθ e d e d e + e e e d 5. e e d ln ln d d 6. d d d d d d + 5 ln ( ln9 ln ) ln 9 9 d 7. sin cos d cos d sin d 8. d ln ln d d d ln ln d ln ln ln ln ln ln ln ln ln( ln ) ln( ln) ln ln

3 9. e ln + e d d + ln e + e ( ) d d tan tan tan tan e d e d. cos 5 sin d 6 cos d sin d cos 5 sin d 5 d cos θ dθ θ d dθ. f ( ) if 5 if < cos θ dθ cos d f ( )d d + 5 d sin sin sin f ( ) cos sin on [,]. g( ) e on [,5] f avg cos sin d sin d cos d g avg 5 5 e d d d d g avg 5 e d 8 5 e d 5 8 e ( 8 e 5 e )

4 5. G ( ) G avg on, ( + ) ( + ) d + ( ) d 5 d 5 [ ] 6. h( ) + d d h avg 5 d h avg 8 on, ( + ) [ ] + + d d 7 d 8 ( ) d ( + ) d 7 d ( )

5 Section. The Fndamental Theorem of Calcls Part Soltions. 5 d ( 5 )d ( ) d DNE becase of a 5 7 e d 6 e d 7 7 eln 7 + eln eln t dt t dt t ( ) ( ) csc ycot ydy csc y csc + csc + 7. sec ydy tan y tan tan 8. 9 z dz 9 z dz ln z 9 [ ln9 ln] ln9 ln7 9. F ( ) ( ) on [,7] F avg 7 7 ( ) d. H ( ) on [,] H avg d ( ) 7 6 9

6 . F F avg ( ) sec on, sec d. F ( ) on [,] F avg d ln [ ln ln] ln. F F ( ) f t ( ) f ( )dt, f ( t) ( ) F ( ) f ( ) t + d, find F ( ) f ( ) + d f ( ) + ( ) F ( ) f ( ) 57. T avg 5. L avg 9 ( 6 + 9e.5t )dt 5.75 F 9 t cos dt 8. hors 8 6. T avg sin t dt 6.5 F 7. g y g y y ( ) t sint dt ( ) y sin y 8. g g ( ) + t dt ( ) + 9. F ( )dt ( ) cos t F ( )dt F ( ) cos t ( ) ( ) cos. h ( ) arctant dt ( ) arctan arctan h. y cost dt y cos t cos

7 Section. Contet for Definite Integrals: Area, Displacement, and Net Change Soltions. y sin,.,. y 8, [,] A. sin d d d A 8 d 8 d d sin sin d d.. sin d cos.. cos + cos. ( ) 8 8 d 6 d y sin cos, [,]. y e, [,.5] A sin cos d cos d sin d A.5 e d d d d.5 e d.75 e d ( ) 5 e.75 ( e.75 ) 5. Displacement ( t 5)dt t 5t 7 5 m 5 Total Distance Traveled ( t 5) dt ( t 5)dt + t 5 5 ( )dt 5 t + 5t + t 5t 5 6 m

8 6. Displacement 6 ( t t 8)dt t t 8t Total Displacement t + t + 8t + t t 8t m 6 t t 8 dt t 6 ( t 8)dt + ( t t 8)dt m

9 Section. Graphical Analysis II Soltions -5. See AP Central

10 Section.5 Accmlation of Rates Soltions. See AP Central for Rbric. See AP Central for Rbric. See AP Central for Rbric. See AP Central for Rbric

11 Section.6 Approimate Integration - Riemann Sms Soltions. Distance Traveled f ( ) + f ( ) + f ( 5) + f ( 7) + f ( 9) Distance Traveled 9 [ ] 9. (a) Right Hand Rectangles, n 8 F avg ( ) 8 F ( )d 8 8 F ( ) + F ( ) + F ( ) + F ( ) + F ( 5) + F ( 6) + F ( 7) + F ( 8) [ ] (b) Left Hand Rectangles, n 8 F avg ( ) 8 F ( )d 8 8 F ( ) + F ( ) + F ( ) + F ( ) + F ( ) + F ( 5) + F ( 6) + F ( 7) [ ] (c) Midpoint Rectangles, n F avg ( ) 8 F ( )d 8 ( ) + F ( ) + F ( 5) + F ( 7) 8 F 8 [ ]. (a) Trapezoidal Rle, n 8 + d 8 f ( ) + f + f + f + f ( ) + f 5 + f 7 + f + f ( ) ( ) + (.57) + (.8) + (.89) + (.65) + (.) + (.) +.95 [ 8 9. ].

12 (b) Midpoint Rle, n 8 + d f 8 + f f f + f f + f + f (a) Trapezoidal Rle, n ln + d f ( ) + f 6 + f f + f + f f f + f f + f ( ) ( ) + (.8) + (.) + (.) + (.6) + (.8) + (.97) + (.) + (.) +. (b) Midpoint Rle, n ln + d f f + f + f + f + f + f + f + f + f [ ] See AP Central for Rbric 6. w ( t)dt represents the change of a child s weight (in ponds) from the age of 5 to r ( t)dt represents how mch oil from the tank (in gallons) has leaked ot in the first mintes. 8. The nits for f ( )d are newton meters, or joles.