Chapter At each point (x, y) on the curve, y satisfies the condition

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Rolf Houston
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1 Chaptr 6. At ach poit (, y) o th curv, y satisfis th coditio d y 6; th li y = 5 is tagt to th curv at th poit whr =. I Erciss -6, valuat th itgral ivolvig si ad cosi.. cos si. si 5 cos 5. si cos 5. cos 6. si 8. y' ta sc Dirctio Filds I Ercis 9, a diffrtial quatio, a poit, ad a blak dirctio fild (a) Sktch th approimat solutios of th diffrtial quatio o th dirctio fild, o of which passs through th idicatd poit. (b) Us itgratio to fid th particular solutio of th diffrtial quatio ad us a graphig utility to graph th solutio. Compar th rsult with th sktchs i part (a). dr 9. si,(,) I Erciss 7 ad 8, vrify Wallis s Formulas by valuatig th / 7. cos / 7 8. cos 6 5 I Erciss 9-6, valuat th itgral ivolvig scat ad tagt.. sc 5. sc 9. sc. ta 5 sc ta.. ta sc 5. sc 6 ta 6. sc ta I Erciss 7 ad 8, solv th diffrtial quatio. I rciss ad, valuat th. si cos. si si d I rciss -6, valuat th Us a symbolic itgratio utility to cofirm your rsult.. cot. csc d cot. t dt csct 5. sc ta 6. ta t sc tdt I Erciss 7-, valuat th dfiit Us a graphig utility to cofirm your rsult. dr 7. si d

2 7. si 8. ta cost 9. dt si t. cos I Erciss -, us a symbolic itgratio utility to valuat th Graph th atidrivativs for two diffrt valus of th costat of itgratio.. cos sc 5.. sc 5 ta I Erciss ad 5, us a symbolic itgratio utility to valuat th. si d 8. y si, y, I Erciss 9 ad, us itgratio by parts to vrify th rductio formula. 9. si cos si si. cos m si m cos si m cos si m I Erciss -, us th rsults of Erciss ad, to valuat th 5. si. sc 5 cos t dt. Match th atidrivativ with th corrct (a) 6 5. si (b) 6 I Ercis 6, (a) fid th idfiit itgral i two diffrt ways. (b) Us a graphig utility to graph th atidrivativ (without th costat of itgratio) obtaid by ach mthod to show that th rsults diffr oly by a costat. (c) Vrify aalytically that th rsults diffr oly by a costat. 6. sc ta 7. Ara Fid th ara of th rgio boudd by th graphs of th quatios y si, y,,. Volum ad Ctroid I Ercis 8, for th rgio boudd by th graphs of th quatios, fid (a) th volum of th solid formd by rvolvig th rgio about th -ais ad (b) th ctroid of th rgio. (c) (d) l 6 6 C 6. 8l 6 6 C arcsi C

3 ( ) arcsi C Evaluat th idfiit itgral usig th substitutio 5si. 9. (5 ) / 5 5. Evaluat th idfiit itgral usig th substitutio sc Evaluat th idfiit itgral usig th substitutio ( ) Evaluat th ta / ( ) arc sc( ) Complt th squar ad valuat th Evaluat th itgral usig (a) th giv itgratio limits ad (b) th limits obtaid by trigoomtric substitutio. / t 68. dt / ( t ) Us a symbolic itgratio utility to valuat th Vrify th rsult by diffrtiatio

4 Evaluat th itgral by makig th giv substitutio. 7. cos() u 7. u 7. ( ) u Evaluat th idfiit 75. ( ) (l ) dt ( t ) 6 8. si( ) d cos si 8. cot csc sc ta 87. Evaluat th idfiit Illustrat ad chck that your aswr is rasoabl by graphig both th fuctio ad its atidrivativ (C=). 88. ( ) 89. si cos Evaluat th dfiit 5 9. ( ) 5 9. ( ) 9. cos( t) dt ( ) / ta d / l Evaluat th itgral usig itgratio by parts with th idicatd choics of u ad dv. 98. l u l dv Evaluat th 99.

5 . si( ). cos(). (l ). r l rdr. si( )d 5. t t dt / 6. cos() / 7. si 8. l / 9. cos l(si ) d / 6 First mak a substitutio ad th us itgratio by parts to valuat th. si. / cos( ) d Evaluat th idfiit Illustrat, ad chck that your aswr is rasoabl, by graphig both th fuctio ad its atidrivativ (C=).. cos( ). ( )

+ x. x 2x. 12. dx. 24. dx + 1)

+ x. x 2x. 12. dx. 24. dx + 1) INTEGRATION of FUNCTION of ONE VARIABLE INDEFINITE INTEGRAL Fidig th idfiit itgrals Rductio to basic itgrals, usig th rul f ( ) f ( ) d =... ( ). ( )d. d. d ( ). d. d. d 7. d 8. d 9. d. d. d. d 9. d 9.