6.5  Areas of Surfaces of Revolution and the Theorems of Pappus


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1 Lecture_06_05.n Ares of Surfces of Revolution n the Theorems of Pppus Introuction Suppose we rotte some curve out line to otin surfce, we cn use efinite integrl to clculte the re of the surfce. Defining Surfce Are The simplest cse to consier is cyliner, the surfce re is S = 2 p rh. Consier frustrum, which is otine y rotting line segment (tht is not prllel to the xis of rottion) out some line.
2 Lecture_06_05.n 2 The surfce re of frustrum is S = 2 p r 1 + r ÅÅÅÅÅÅÅÅÅÅÅÅÅ 2 h = phr r 2 L h where r 1 is inner rius, r 2 is the outer rius, n h is slnt length. Now, consier the soli otine rotting curve C out some line. For the ske of simplicity, consier the curve y = f HxL etween x = n x = tht is rotte out the xxis. y x As efore, prtition the D to otin x 0 =, x 1, x 2,..., x n =. We will use frustrum to pproximte the surfce re of ech segment. Consier the kth segment, etween x k1 n x k. The surfce re is f Hx Frustrum surfce re = 2 p ÿ k1 L + f Hx k L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ "################ HD x ############### 2 k L 2 + HD y k L 2 = pÿh f Hx k1 L + f Hx k LL "################ HD x ############### k L 2 + HD y k L 2 = pÿh f Hx k1 L + f Hx k LL $%%%%%%%%%%%%%%%% HD x k L 2 J1 %%%%%%%%%%%%%%%% + I ÅÅÅÅÅÅÅÅ D y k %%%%%%% D x k M 2 N Since f is ifferentile D, it is ifferentile on every suintervl D, thus y the Men Vlue Theorem, there is numer c k k1, x k D such tht Thus, f Hx kl  f Hx k1 L x k  x k1 = D y k ÅÅÅÅÅÅÅÅ D x k f ' Hc k L = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Frustrum surfce re = pÿh f Hx k1 L + f Hx k LL "################ 1 + H f ' Hc ########## k LL 2 D x k The pproximtion of the totl surfce re is given y n S º pÿh f Hxk1 L + f Hx k LL "################ 1 + H f ' Hc ########## k LL 2 D x k k=1 We cn otin etter n etter pproximtion y letting nø.
3 Lecture_06_05.n 3 DEFINITION Surfce Are for Revolution Aout the xxis If the function f HxL 0 is continuously ifferentile D, the re of the surfce generte y revolving the curve y = f HxL out the xxis is 2 p y $%%%%%%%%%%%%%%%% 1 + I ÅÅÅÅÅÅÅ y %%%% x M2 x = Ÿ 2 p f HxL "################ 1 + H f ' ########## HxLL 2 x ü Exmple Fin the re of the surfce generte y revolving the curve y = è!!! x etween ÅÅÅÅ 3 4 x ÅÅÅÅÅ 15 out the xxis. 4
4 Lecture_06_05.n 4 Revolution Aout the yxis DEFINITION Surfce Are for Revolution Aout the yxis If the function x = ghyl 0 is continuously ifferentile D, the re of the surfce generte y revolving the curve x = ghyl out the yxis is c 2 p x $%%%%%%%%%%%%%%%% 1 + I ÅÅÅÅÅÅÅ x %%%% y M2 y = Ÿ c 2 p ghyl "################ 1 + Hg' HyLL ########## 2 y ü Exmple Fin the re of the surfce generte y revolving the region curve x = y + 2 è!!! y etween 1 y 2 out the yxis.
5 Lecture_06_05.n 5 Prmeterize Surfces We cn lso consier the cse where x n y re given y set of prmetric equtions. If the curve is prmeterize y the set of equtions x = f HtL n y = ghtl, t, where f n g re continuously ifferentile D, then we hve the following. Surfce Are for Revolution for Prmeterize Curves If smooth curve x = f HtL, y = ghtl, t, is trverse exctly once s t increses from to, then the res of the surfces generte y revolving the curve out the coorinte xes re s follows: 1. Revolution out the xxis (y 0) 2 p y $%%%%%%%%%%%%%%%% H ÅÅÅÅÅÅÅ x %%%%%%%%%%%% t L2 + I ÅÅÅÅÅÅÅ y t M2 t = Ÿ 2 p ghtl "################ H f ' HtLL 2 ################ + Hg' HtLL ##### 2 t 2. Revolution out the yxis (x 0) 2 p x $%%%%%%%%%%%%%%%% H ÅÅÅÅÅÅÅ x %%%%%%%%%%%% t L2 + I ÅÅÅÅÅÅÅ y t M2 t = Ÿ 2 p f HtL "################ H f ' HtLL 2 ################ + Hg' HtLL ##### 2 t ü Exmple Fin the re of the surfce generte y revolving the curve x = ÅÅÅÅ 2 3 t3ê2, y = 2 è!! t, 0 t è!!! 3 out the xxis.
6 Lecture_06_05.n 6 The Differentil Form The equtions 2 p y $%%%%%%%%%%%%%%%% 1 + I ÅÅÅÅÅÅÅ y %%%% x M2 x n re often written in terms of s= "################ H xl 2 + ########## H yl 2 s S = Ÿ 2 p y s n S = Ÿ c 2 p x s c 2 p x $%%%%%%%%%%%%%%%% 1 + I ÅÅÅÅÅÅÅ x %%%% y M2 y In the first integrl, y is the istnce from the xxis to n element of rc length s. Similrly, for the secon integrl, x is the istnce from the yxis to n element of rc length s. Thus, oth integrls re of the form S = Ÿ 2 p HriusL Hn withl = Ÿ 2 p r s where r is the rius from the xis of revolution to n element of rc length s. Cylinricl Versus Conicl Bns Why not use cylinricl inste of conicl (frustrum) ns? If we revolve the curve y = f HxL, x out the xxis, we otin the integrl S = Ÿ 2 p f HxL s This formul fils to give results consistent with the surfce re formuls from geometry, which is wht we wnt (consistency). The Theorems of Pppus THEOREM 1 Pppus's Theorem for Volumes If plne region is revolve once out line in the plne tht oes not cut through the region's interior, then the volume of the soli it genertes is equl to the region's re times the istnce trvele y the region's centroi uring the revolution. If r is the istnce from the xis of revolution to the centroi, then V = 2 p r A
7 Lecture_06_05.n 7 THEOREM 2 Pppus's Theorem for Surfce Ares If n rc of smooth plne curve is revolve once out line in the plne tht oes not cut through the rc's interior, then the re of the surfce generte y the rc equls the length of the rc times the istnce trvele y the rc's centroi uring the revolution. If r is the istnce from the xis of revolution to the centroi, then S = 2 p r L ü
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