FM Applications of Integration 1.Centroid of Area

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1 FM Applicions of Inegrion.Cenroid of Are The cenroid of ody is is geomeric cenre. For n ojec mde of uniform meril, he cenroid coincides wih he poin which he ody cn e suppored in perfecly lnced se ie, is cenre of mss. The cenre of mss for disriuion of weighs in dimensions re mii nd m i y These formuls cn e generlised y using clculus o give he resul elow. m y i m i i Cenroid of wo-dimensionl figures The cenroid (, y) of plne figure y f () ounded y he lines = nd = sisfies: nd where A Ay A y y y y = f() δ Emple Find he coordines of he cenroid of he plne figure enclosed y he curve y =, he - nd y-es nd he line =. Soluion The re of he figure is given y y ( ) So, using A y we oin y Using Ay y gives A 6 8 y y Hence, he coordines of he cenroid re, Emple Find he coordines of he cenroid of he plne figure enclosed y he curve y =, he -is nd he line 9 =. [Ans:, ) ] ( 5 FM Pure Applicions of Inegrion 8//5

2 Emple Clcule he cenroid of he ringle wih verices (, h), (, ) nd (, ). [Ans: (/, h/)] Emple Find he coordines of he cenroid of he uniform plne figure enclosed y he curves y = nd y = nd he y-is. Soluion y = y = = The required figure is he region elow he curve y = wih he region elow he curve y = removed The required poin of inersecion of he curves is given y = = Using he nswers from Egs nd, nd using, y for he cenroids of he required regions, le cn e formed using he mehods from mechnics y = Required shpe Are 8 -coordine y-coordine 9 y y = To find -coordine 8 To find y-coordine 8 y 9 y 8 5 FM Pure Applicions of Inegrion 8//5

3 FM Applicions of Inegrion.Cenroid of Volume Cenroids of solids of revoluion A solid of revoluion is found y roing pr of curve y = f() hrough 6 round he -is. The volume of solid so formed is symmery) nd hve -coordine sisfying: y. Is cenroid, or geomeric cenre, will lie on he -is (y V y where V y δ nd y Emple The re enclosed y he curve y = +, he - nd y-es nd he line = is roed ou he -is hrough one complee revoluion. Show h he cenroid of he solid formed hs -coordine.. Soluion y 6 9 V y 9 V 6 5 V y V 6 Hence, Sndrd resuls The formuls for he cenroid of solid hemisphere nd solid cone or pyrmid pper in he CIE formul shee. Cenre of mss of solid hemisphere, rdius r: r from cenre 8 Cenre of mss of solid cone or pyrmid of heigh h: h from vere FM Pure Applicions of Inegrion 8//5

4 Eercise Q. Clcule he cenroids of re for he following curves eween he limis given: (i) y from = o = (ii) y from = o = (iii) y cos from = o = (iv) y e from = o = Q. Find he cenroid of he re enclosed y he curves y = nd y = Q. Find he cenroids of he volumes formed when he regions ounded y he following curves nd he -is re roed hrough one complee revoluion ou he -is: (i) (ii) y y sin from = o = π (iii) y from = o = (iv) (v) y from = o = y from = o = (vi) y from = o = (vii) y e from = o = (viii) y e from = o = Answers Q (i) (/75, 5/5) (ii) (9/6, 7/) (iii) (pi/, pi/8) (iv) (/(e ), (e + )/) Q. (/, /5) Q (i) (, ) (ii) (pi/, ) (iii) (7/8, ) (iv) (5/8, ) (v) ( ln, ) (vi) (9/7, ) (vii) ((+e )/(e ), ) (viii) ((e )/(e ), ) FM Pure Applicions of Inegrion 8//5

5 FM Applicions of Inegrion.Arc Lengh Along A Curve Arc lengh long curve Given curve wih equion y = f() he disnce, S, long he curve from = o = is given y: L dy Arc lengh, L y = f() (See Guler & Guler p.5) Emple Show h he lengh of he rc of he curve = y from = o = is [Noe: You cn use he susiuion u Soluion 9. or u o inegre ] Arc lengh long prmeric curve A curve my e epressed in erms of prmeer in he form = f() nd y = g(). According o he chin rule: dy dy so dy dy dy Hence dy dy nd dy dy Thus, he rc lengh long prmeric curve is given y: dy L FM Pure Applicions of Inegrion 5 8//5

6 Emple A curve is given prmericlly y is e. e cos nd y e sin for. Show h he lengh of he curve Soluion dy e cos e sin e sin e cos dy e cos sin e L e e e Arc lengh long polr curve s required. If he curve is given in polr form, r = f(θ) hen he rc lengh is L r dr d d Emple Show h he lengh of he circumference of he crdioid cos r is 8. Emple 5 Find he lengh of he rc of he polr curve r e for where o [Ans: () 8 () (e - ) ] Eercise Guler & Guler p.55 Eercise C Q,,, 8,, 6(i), 7(, i) FM Pure Applicions of Inegrion 6 8//5

7 FM Applicions of Inegrion.Curved Surfce Are of Revoluion Curved surfce re of revoluion If he rc of curve y = f() from = o = is roed hrough 6 round he -is, hen he surfce re of he shpe formed is given y: y = f() S y dy (See Guler & Guler p.5) Emple The rc of he curve y = eween = nd = is roed hrough rdins ou he -is. Show h he vlue of he surfce re genered is ( ). 7 Curved surfce re of revoluion for prmeric curve Using he sme rgumen s ove, he surfce re of curved surfce of revoluion of prmeric curve is dy given y: S y Emple The prmeric equions of curve re ( sin ) nd y ( cos) where is posiive consn. dy Show h sin. The rc of he curve eween = nd = is roed compleely ou he -is. Show h he re of he surfce of revoluion formed is given y cos sin 8 nd hence find his re. [Ans: 6 /] Eercise Guler & Guler p.55 Eercise C Q6( d, f),,,, 6(ii), 7(ii), 8, 9, FM Pure Applicions of Inegrion 7 8//5

8 FM Applicions of Inegrion 5.Men Vlue Theorem The re under he curve y = f() from = o = is given y he inegrl f ( ). The lengh of he inervl is, of course,, so i follows h he men vlue of f() over his inervl sisfies: men vlue f ( ) Hence, he men vlue of f() over he inervl [, ] = f ( ) Emples Find he men vlues of he following funcions over he given regions:. sin θ over he inervl. ( )( ) for [Ans /]. for [Ans -]. cos for [Ans ln ] 5. for [Ans /] 6. e for [Ans ln (/)] [Ans e ] FM Pure Applicions of Inegrion 8 8//5

9 FM Applicions of Inegrion 6.Older Em Quesions Winer [6 /5] Winer Summer Summer [88] FM Pure Applicions of Inegrion 9 8//5

10 Summer Q Eiher [((e 5)/(e ), (e 5)/8e(e ))] Winer Q Or [/, (5 )/8] Summer FM Pure Applicions of Inegrion 8//5