Generalized Surface Area of Revolution

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1 Generlized Surfce Are of Revolution Richrd Winton, Ph.D. Michel. Wrren Astrct Suose curve in the lne R is defined y continuous function over closed ounded intervl. A forul is develoed for the rdius of revolution fro nonverticl liner is of revolution to. An lternte derivtion is lso rovided. The rdius of revolution is then used to roduce forul for the surfce re generted y revolving out. This result is coined with the stndrd forul for surfce re out verticl is to yield generlized forul for the surfce re generted y revolving out n ritrry liner is of revolution. Introduction Mny lictions of derivtives nd integrls re routinely studied in clculus. These lictions cn soeties e etended to ore generl setting thn is norlly found in the clculus tetooks, such s the result on centroids in [9]. Another concet coonly studied in clculus is tht of the surfce re generted y revolving continuous curve out line in the lne R. Soe tetooks liit this suject to the revolution of curves out the nd y es ([],[],[5],[6],[7],[8]). In these cses, if curve is defined y y f(),, then the surfce re generted is SA f () ds f () f () d () when is revolved out the -is nd SA ds f () d () when is revolved out the y-is, where ds f () d is the differentil rclength. Other tetooks, however, include the soewht ore generl cses of revolving curves out ritrry horizontl nd verticl lines in R ([3],[4]). In these ore generl cses, the surfce re is given y Journl of Mtheticl Sciences & Mthetics Eduction, Vol. 5 No.

2 f () t f () SA f () t ds when is revolved out the horizontl line y t nd d (3) SA t ds t f () d (4) when is revolved out the verticl line t. The gol of this er is to develo forul for the surfce re roduced y revolving continuous curve out coletely ritrry line in R. Since verticl lines re not functions, then the surfce re roduced y revolving curve out verticl is of revolution rovided in (4) ove ust e considered sertely. Thus the secific gol here is to develo forul for the surfce re generted y revolving continuous curve out n ritrry nonverticl line, gretly generlizing (3) ove. The resulting forul, together with (4), will then rovide the result sought. Rdius of Revolution In cses () nd (3) ove, the rdius of revolution r of oint P reltive to horizontl is of revolution is the verticl distnce etween P nd. In siilr nner, in cses () nd (4) ove, r is the horizontl distnce etween the oint P nd the verticl is of revolution. In ll of the ove cses, r cn e descried s the length of the unique line segent T in R with the following roerties: () One endoint of T is P. () The other endoint of T lies on. (c) T is erendiculr to. Using this generl descrition for r, ll four of the ove cses cn e condensed into the single forul SA r ds r f () d (5) The gol of this er then reduces to deterining ore generl forul for r for ll nonverticl es of revolution, which includes the forul in (3) reltive to horizontl lines s secil cse. To this end, suose curve is defined y continuous function y f() for. Suose further tht the is of revolution is defined y the liner function A() t, where nd t re rel nuers nd 0. (See Figure.) Journl of Mtheticl Sciences & Mthetics Eduction, Vol. 5 No.

3 y y f() y A() Figure If, then the oint on corresonding to is P(,f()). (See Figure.) y P y f() y A() Figure Journl of Mtheticl Sciences & Mthetics Eduction, Vol. 5 No.

4 The sloe of the line through P nd erendiculr to is. Thus the eqution of is y f() ( ), or y f () ( ). (See Figure 3.) y P y f() y A() Figure 3 To deterine the oint of intersection Q of with, we set t f () ( ). Therefore t f (), so tht f () t. Thus f () t f() t, nd so. Sustituting this eression for into y t yields f () t f () t t( ) y t f () t t t f () t. Thus Q hs coordintes f () t f () t Q,. (See Figure 4.) Journl of Mtheticl Sciences & Mthetics Eduction, Vol. 5 No.

5 y P y f() y A() Q Figure 4 The rdius of revolution r of the oint P out the is is therefore the length of the segent T with endoints P nd Q. Using the distnce forul in R, we hve r d(p,q) f () t f () t f () f () t ( ) f () t f ()( ) f () t f () t f () f () f () t t f () Journl of Mtheticl Sciences & Mthetics Eduction, Vol. 5 No.

6 f () t ( ) f () t f () t ( ) f () t f () ( t) f () A(). (6) Alternte Derivtion of the Rdius of Revolution A different, nd erhs soewht less intuitive, derivtion of (6) is found in [7, ]. For this lternte roch, suose is the cute ngle etween the -is nd the is of revolution. Then tn () nd tn(), so tht cos(). (See Figure 5.) Figure 5 If R is the oint on verticlly ove or elow the oint P(,f()), then R hs coordintes R(,A()). Thus in tringle PQR we hve d(p,r) f () A(). Furtherore, QPR is congruent to since is erendiculr Journl of Mtheticl Sciences & Mthetics Eduction, Vol. 5 No.

7 to. (See Figure 6.) y P y f() y A() r Q R Figure 6 Hence r f () A() f () A() cos(), nd so r f () A() cos() f () A(), which is consistent with (6). Surfce Are We re now rered to generlize forul (3) to include ll nonverticl, nonhorizontl es of revolution. Alying (6) to ech vlue of for, the rdius of revolution of the oint (,f()) on the curve out the is is r() f () A(). Hence the surfce re generted y revolving out is SA r()ds f () f () A() d Journl of Mtheticl Sciences & Mthetics Eduction, Vol. 5 No.

8 f () A() f () d, (7) where f () A() is the verticl distnce etween nd for ech such tht. Note, however, tht when the is of revolution is horizontl, then 0. In this cse the eqution of silifies to A() t. onsequently, (7) reduces to SA f () t f () d, which is consistent with (3) ove. Hence the cse for horizontl es of revolution when 0 is included in (7). onclusion oining (4) with (7), we hve the following conclusion which includes ll liner es of revolution in R. If curve is defined y continuous function y f() for, then the surfce re generted y revolving out liner is of revolution is SA t f () f () A() d f () d if is verticl defined y if is defined y A() t t. Richrd Winton, Ph.D., Trleton Stte University, Tes, USA Michel. Wrren, Trleton Stte University, Tes, USA References [] Howrd Anton, Irl Bivens, nd Stehen Dvis, lculus, 7 th ed., John Wiley & Sons, Inc., New York, 00. [] Ross. Finney nd George B. Thos, Addison-Wesley, New York, 990. [3] John B. Frleigh, lculus with Anlytic Geoetry, nd ed., Addison- Wesley, Reding, Msschusetts, 985. [4] Rolnd E. rson, Roert P. Hostetler, nd Bruce H. Edwrds, lculus with Anlytic Geoetry, 4 th ed., D.. Heth nd ony, eington, Msschusetts, 990. [5] Ae Mizrhi nd Michel Sullivn, 3 rd ed., Wdsworth, Belont, liforni, 990. [6] S.. Sls nd Einr Hille, lculus: One nd Severl Vriles, 6 th ed., John Wiley & Sons, Inc., New York, 990. [7] Jes Stewrt, lculus, 5 th ed., Thoson, United Sttes, 003. Journl of Mtheticl Sciences & Mthetics Eduction, Vol. 5 No.

9 [8] Erl W. Swokowski, Michel Olinick, nd Dennis Pence, lculus of Single Vrile, nd ed., PWS Pulishing, Boston, 994. [9] Richrd Winton, Syetry t Infinity, ollege Mthetics Journl 36 (005) 8-3. Journl of Mtheticl Sciences & Mthetics Eduction, Vol. 5 No.

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