A 2 ab bc ca. Surface areas of basic solids Cube of side a. Sphere of radius r. Cuboid. Torus, with a circular cross section of radius r
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1 Sufce e f ic lid Cue f ide R See f diu 6 Cuid c c Elliticl cectin c Cylinde, wit diu nd eigt Tu, wit cicul c ectin f diu R R Futum, ( tuncted ymid) f e eimete, t e eimete nd lnt eigt. nd e te eective e nd t e im, f lengt wit unifm c ectinl e wit eimete ume ll lnt e te me lengt Otewie we ve n lique futum ymid nd tee fmule will nt ld ymid, wit e eimete nd lnt eigt Cne, wit e diu nd lnt eigt Ti i te cuved ufce e i.e. n en cne Mtemtic tic ndut: e D ndew Fenc. GE
2 Clculting te ufce e f (igt) ymid i.e. ll lnt e te me lengt Otewie we ve n lique ymid nd ti fmul de nt ld n n lnt eigt n 0 Teefe te ttl etei f te ymid i te um f te etei e f te lmine w te um f te fit intege i... ( ) e f igt ectngul ed ymid eimete n ( ) e f igt cne ymid f eendicul lnt eigt nd e eimete cn e tugt f eing cmed f n (infinitely) lge nume f imil lmine. Te enlgement fct f ec lmin i in diect tin t te lnt ditnce fm te e f te ymid. Ti i tue ince te ide f te ymid e tigt line wit cntnt gdient. Teefe ecme infinite, te e tend twd dd te e e t mke te ttl ufce e i.e. te ttl e f ymid i te e lu lf te duct f te e eimete nd te lnt eigt Let tee e lmine f tickne /. Since ll lmine e imil, te eimete cle n/ wee n i te lmin nume fm te e. Te etei e f ec lmin i: n n n n n n te we cn find te cuved e f igt cne y nting w it cn e cntucted fm cicle wit ect miing c lengt Mtemtic tic ndut: e D ndew Fenc. GE
3 Clculting te e f ee witut uing clculu S we cn cnclude tt te ti e, in te limit f mll, de nt deend n ngle. i.e. ec ti n te emiee m (i.e. cn e jected nt) cylindicl ti f (mll) widt. Ti men te cuved ufce f te emiee mut equl tt f te encling cylinde. Hence te cuved ufce e f te emiee mut e: c Teefe te ufce e f ee i: Cnide te tngent t cicul c ectin f emiee t int, wic i t elevtin ngle fm te e f emiee f diu. int i ditnce wy lng te tngent uc tt c, wee i te jected widt f te emiee n cylinde f diu. (Te cylinde widt etween int t te eigt f itin nd ). w if we ink, ten itin will mve cle t te cicle, nd ence in ti limit we wuld eect te c lengt etween nd t ecme: c Te e f cicul ti wit widt etween nd n te emiee i imtely: c c c Tke n nge. Dw und it fu time. eel te nge. Te eel uld fit lmt ectly int te fu cicle, ince te e f ec cicle i Mtemtic tic ndut: e D ndew Fenc. GE 3
4 futum i tuncted ymid Cuved e f entie ymid (ume OT lique) entie l ( ) Clculting te e f ee uing clculu te ll ngle e in din i.e. din = 80. Since c i imil t entie ymid, cuved f emved c ymid i: Teefe futum cuved e i l l emved enite w ince te t nd e f te futum e imil l l l l l l l l l ( ) l emved l l d in Divide u te ufce f emiee int cicul in Te diu f ec in i in nd te widt i te c lengt d Te e f ec in i d in d in d Te ttl e f te ee i teefe 0 0 in d c c c(0) 0 d Teefe ttl ufce e f te futum i: Ti i nly tue f nn lique futum! Mtemtic tic ndut: e D ndew Fenc. GE 4
5 cimede metd f clculting te e f ee Fitly ti invlve finding fmul f te ufce e f vlume f evlutin f egul ctgn ut te i. Genelizing f egul even-nume ided lygn f inceing lge nume f ide, we nticite te eulting ufce e t e tt f ee. H H C G D F E ting te llel line cnnecting vetice nd HC, nd y lictin f te Z-ngle teem, ne cn w tt te geen igt ngled tingle igligted e imil 3 E k k k k G k D F Since we cn cicumcie te ctgn wit cicle, we cn ue te igt ngle dimete teem t w te nge cicle i l imil t te geen tingle. Define k = E vlume f evlutin f te ctgn i te vlume f tw cne fmed y te ttin f tingle H nd tw futum fmed y te ttin f tezium HGC Te ttl ufce e i teefe: Uing te eult ve k k 4 ke w we wuld eect te me eult f ny even lygn, ut te nume f ide incee, we eect k t c E ink F cicle E =. Teefe if in ti ceni k = E 00 ided lygn f evlutin ti lk vey muc like ee! 0 ided lygn f evlutin Mtemtic tic ndut: e D ndew Fenc. GE 5
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