Calculus 2 Quiz 1 Review / Fall 2011

Transcription

1 Calcls Qiz Review / Fall 0 () The fctio is f a the iterval is [, ]. Here are two formlas yo may ee. ( ) ( ) ( ) 6 (a.) Use a left-e, right-e, a mipoit sm of "" rectagles to approimate. The withs of all rectagles is the same, amely: w, bt their heights vary. Let "h l ", "h r ", a "h m " be the heights of the " th " rectagle for the left-e, right-e, a mipoit techiqes. Here is how those respective heights vary. h l f l f ( ) ( ) h r f r f h m f m f Page of 7

2 The proct of the with a the sm of the heights of all "" rectagles yiels a approimate vale for provie that the sig is tae ito accot for the height. Let "A L ", "A R ", a "A M " be the respective vales for the left-e, right-e, a mipoit approimatio techiqes. A L A l A R A r A M A m Sice "" oes ot epe o "", we ca rewrite the three epressios above as follows. A L A l A R A r A M A m Page of 7

3 (b). Determie the EXACT vale of sig each of three area approimatio techiqes by lettig "" approach ifiity for the reslts that yo obtaie i part (a.). Oly the right-e area approimatio is oe here; bt yo shol try to o liewise for oe or both of the others. A R A R A M A R A R 0 A R 00 We ca se two of the give formlas at the top of page # to replace the two smmatios i the above eqatio. ( ) ( ) ( ) 6 Page of 7

4 A R 00 ( ) [ ( ) ( ) ] 6 A R A R (c). Determie the vale of by evalatig the efiite itegral sig the Fametal Theorem of Calcls to evalate. Page of 7

5 (). Use five () rectagles to approimate techiqe. sig the right epoit 00 A R ( ) [ ( ) ( ) ] 6 00 A R ( ) [ ( ) [ ] ] 6 A R (e). O the graph below, show the five () rectagles for the right-e techiqe to approimate. Rectagle Right Epoit Estimate y 0 Page of 7

6 (f).use appropriate formla(s) from geometry to approimate The graph of f loos lie this.. f() over give iterval y 0 We ca approimate sig three triagles as follows. the first triagle lies etirely below the -ais a covers the sbiterval: [, 0 ]. The seco triagle also lies etirely below the -ais a it covers the sbiterval: [0, ]. Thir triagle covers the sbiterval: [, ] a lies etirely above the -ais. Let "A" be the approimate et "sige area" a "A ", "A " a "A " be the "sige areas", respecively, of the first, seco, a thir triagles. A A A A A A A A Page 6 of 7

7 (). Cosier the regio boe o the left by the lie y, a o the right by the crve y. The figre below shows their graphs. y 0 (a). Fi the two poits of itersectio of the two give crves that apply to the specifie regio. y y y y y y ( y ) ( y ) 0 Ths, these are the two poits: ( ), a ( ). Page 7 of 7

8 (b). Write a efiite itegral or itegrals that eactly eqals the area of the boe regio. Itegrate with respect to "y". DO NOT EVALUATE THE INTEGRAL(S)! Horizotal Rectagles/ Itegral y 0 Use horizotal rectagles. R y L y A R L y y ( y) y y y y A y y y (c). Write a efiite itegral or itegrals that eactly eqals the area of the boe regio. Itegrate with respect to "". DO NOT EVALUATE THE INTEGRAL(S)! Page 8 of 7

9 Vertical Rectagles/ Itegrals y 0 Use vertical rectagles. Therefore, solve for " y " i terms of " ". y y 0 y y y The top crve is the pper brach of the qaratic bt the bottom crve to the left of is the straight lie whereas the bottom crve is the lower brach of the qaratic to the right of. For that reaso, we ee itegrals to calclate the eclose area. Page of 7

10 A ( ) A A (.) Evalate the followig iefiite itegrals. (a) C (b). 7 e si cos 7 e si cos 7l e cos si ta si C Page 0 of 7

11 (c). csccsc csccot csc csc cot ( csc ) cot C csccsc csccot cotcsc C (). e e 6 e e 6 e e e e e e 6 cot C e e 6 cot e C Page of 7

12 (.) Evalate the followig efiite itegrals. (a). cos cos cos 6 (b). 6 cot si cos 6 cos si l si 6 si 6 cot l l l Page of 7

13 (c). t t t t t t t t 0 l 8 t t t 0 8 l 0 8 l 0 ( l( 8) ) 0 l 8 l (). l ( ) ( 7 8) 8 Page of 7

14 (e). l Page of 7

15 (f). e ( l ) l l e l( e) l l l l l l e ( l ) l Page of 7

16 (.) Use the appropriate formla(s) from geometry to evalate the itegrals. (a). ( ) y Itegra 0 0 The triaglar area above the -ais has as its base the sbiterval: [, ] a has a height of "". Let "A " eqal the "sige area" of that triagle. The area below the -ais cosists of two eqal triaglar areas. Oe has as its base the sbiterval: [, ], a the other has as its base the sbiterval: [, ]. The heights of both of those triagles is "". Let Let "A " eqal the "sige area" of oe of those two triagles. ( ) A A Page 6 of 7

17 (b). y Itegra 0 The first itegral o the right sie of the above eqatio eqals the rectaglar area with a base of "6" its a a height of "" its. The area is ths "". The seco itegral o the right sie of the above eqatio eqals the semicirclar area with a rais of "" its. The area is ths " ( ) ". Page 7 of 7