MTH 146 Class 11 Notes


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1 8. Are of Surfce of Revoluion MTH 6 Clss Noes Suppose we wish o revolve curve C round n is nd find he surfce re of he resuling solid. Suppose f( ) is nonnegive funcion wih coninuous firs derivive on he inervl,. Also suppose h curve C is defined y he equion y f ( ) where. We pproime C in he sme wy we pproimed i in secion 8.. Then, o pproime he desired surfce re, we revolve he pproime version of C round he is. A picure represening he siuion is elow (keep in mind h ech srigh line segmen h you see will e revolved round he is). Noe: I is imporn o noe h in his secion we re ssuming f is nonnegive funcion. Quesion: Wh kind of shpe is creed if he segmen highlighed in rown is revolved round he is?
2 Revolving ech of he segmens h pproime he curve round he is crees n nds ech of which we cn find he re of using our knowledge of lerl surfce re of cones (he lerl surfce re of cone is rl where l is he cone s sln heigh). Specificlly, ech nd is he surfce of frusum ( picure of frusum is elow). We will ssume for he momen h none of our nds is cylinder. Suppose r is he rdius of he righ circle, r is he rdius of he lef circle, l is he sln heigh of he frusum, nd l is he sln heigh of he righ circulr cone wih he righ circle s is se. Then, he surfce re of r l l rl r r l r l. he frusum is This formul cn e simplified y noing h from similr righ ringles wihin he cone r l conining he frusum we oin: rl rl r l rl r r l. Thus, he r l l formul for he surfce re of he frusum ecomes r r l rr l. Noe: I is possile h he lef nd righ circle re ecly he sme nd we cully hve cylinder. One cn noe h he re formul h we oined works for cylinder s well. So, he re of he nd formed y revolving he rown segmen (in he ove picure) round f i f i he is is PP i i. This mens h our pproimion for he surfce re is: f f n i i Pi Pi i. Now, from 8. we know h here is Pi Pi f i i i nd f i f i f f n i in, i i h is such h i. Furhermore for smll we know h since f is coninuous. So, noher pproimion for he surfce re is: f i f i. This pproimion ges eer nd eer s n increses nd if
3 we ke he limi of his sum s n we recognize i s definie inegrl! This leds o he following definiion. Definiion: Suppose f( ) is nonnegive nd hs coninuous derivive on,, we define he surfce re of he surfce oined y roing he curve y f ( ) where ou he is s: dy d S f ( ) f ( ) d y d y ds where ds is he rc lengh differenil (see secion 8. for definiion of rc lengh differenil). If he curve is descried s g( y) where c y d hen he formul ecomes: d d S y dy. dy c Also, for roion ou he yis he formul ecomes (ssuming we re revolving y f ( ) ): dy S d ds d. Noe: In his definiion we re no le o pu limis of inegrion on he inegrls involving he rc lengh differenil ecuse we do no know limis on he s. Emple: Find he ec surfce re of he surfce oined y roing he curve where ou he is. y We noe h f ( ),. is nonnegive nd hs coninuous derivive on So, y definiion he desired surfce re is given y: dy S y d 9 d d. Now, we mke he susiuion 9. This implies h () u u, u() 5, nd du 6 d du d. Using his susiuion, we see: d u du u Anoher wy o do his emple would e o noice h noher wy o define he curve is y where y 8. So, we could hve compued he surfce re
4 8 d y evluing: S y dy y y dy dy. This inegrl is 9 ougher o evlue u one cn confirm h i evlues o Grded Emple: Se up u do no evlue n inegrl which yields he ec surfce re of he surfce oined y roing he curve y ln where ou he yis. Soluion: We will use he formul: S ds. Since we hve y in erms of, we use dy ds d. We noe h dy. So, he nswer is: d S d. Noe: I is firly esy eercise o evlue he inegrl in his grded emple. Emple: Suppose h he pr of he uni circle in he firs qudrn is roed round he y is. Find he surfce re of he resuling solid using he mehods of his secion. We know h he equion of he uni circle is given y y. If we solve his equion for y we see h he op hlf of he uni circle is given y y where. This mens h he curve we re ineresed in revolving round he yis is given y y where. So, he desired surfce re is given y (Noe: we oin n improper inegrl): S d lim d. lim d lim d Using simple Clculus susiuion we see This mens: d C.
5 lim d lim lim. Noe: This resul mkes sense since he solid we oin in his emple is he op hlf of he uni sphere nd we know he surfce re of he uni sphere is. Grded Emple: Griel s horn is he solid oined y roing he region R (, y) nd y ou he is, find wheher or no he surfce re is finie. Jusify your nswer (Hin: Use he comprison heorem). Soluion: We re ineresed in surfce re of he solid oined y revolving he curve given y y where ou he is. By definiion he desired surfce re is: dy lim y d lim d lim d d lim d lim d. We noe h for we hve lim d lim ln. Also, we hve h. So, y he comprison heorem: nd we know h he surfce re of Griel s horn is infinie. lim d is divergen Noe: Ineresingly we cn lso prove h he volume of Griel s horn is finie! See prolem 6 of secion 7.8 in he ook for deils regrding his. Recommended Homework: odd; 9
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