mth 11 pplictio: volumes by shells: volume prt iii 17 Volumes of Revolutio: The Shell Method I this sectio we will derive ltertive method clled the shell method for clcultig volumes of revolutio This method will be esier th the disk method for some problems d hrder for others There re lso some problems tht we cot do with the disk method tht become possible with the shell method We will gi use the subdivide d coquer strtegy with Riem sums to derive the pproprite itegrl formul Our objectives re to develop the volume formul for solids of revolutio usig the shell method; to compre d cotrst the shell d disk methods We strt with cotiuous fuctio y = f (x) o [, b] We crete regulr prtitio of [, b] usig itervls d drw the correspodig pproximtig rectgles of equl width Dx I left hlf of Figure we hve drw sigle represettive pproximtig rectgle o the ith subitervl This time we rotte the rectgle bout the y-xis, ot the x-xis (Compre the right hlf of Figure to the right hlf of Figure 11) f (x i ) Dx b y = f (x) f (x i ) Dx b Whe the rectgle i Figure is rotted roud the y-xis, hollow thiwlled cylider clled shell is creted We wt to determie the volume of the solid prt of the cylider (ot the volume of the hollow prt) A cute wy to do this is to tke to tke the shell d slice it verticlly d the uroll it so tht it forms flt slb s i Figure 7 Figure : Left: The regio uder the cotiuous curve y = f (x) o the itervl [, b] d represettive rectgle Right: The solid shell of outer rdius r = x i, height h = f (x i ), d wll width Dx geerted by rottig represettive rectgle bout the y-xis h = f (x i ) h = f (x i ) Dx pr = px i Dx {z } r = x i The resultig slb hs the sme height s the shell, h = f (x i ) d the sme width w = Dx The the legth of the slb is the sme s the circumferece of the cylider Sice the cylider hs (outer) rdius r = x i, the circumferece of the cylider is pr = px i Sice the slb is relly thi rectgulr box, its volume is V = legth width height Usig Figure 7, this trsltes ito Figure 7: Left: The solid shell of outer rdius r = x i, height h = f (x i ), d wll width Dx geerted by rottig the represettive rectgle bout the y-xis Right: The shell cut ope d lid out s slb of legth pr = px i, height h = f (x i ), d width Dx The volume of the shell d slb re equl: V i = pxf(x i )Dx Volume of represettive shell = V i = lwh = px i f (x i )Dx Approximtig the volume of the etire solid by such shells of width Dx d height f (x i ) produces Riem sum Volume of Revolutio px i f (x i )Dx = p x i f (x i )Dx ()
mth 11 pplictio: volumes by shells: volume prt iii 18 As usul, to improve the pproximtio we let the umber of subdivisios! d tke limit Recll from our erlier work with Riem sums, this limit exists becuse xf(x) is cotiuous o [, b] sice x d f (x) re both cotiuous there So Volume of Revolutio by Shells = lim p! x i f (x i )Dx = p xf(x) dx (7) where we hve used the fct tht the limit of Riem sum is defiite itegrl Of course, we could use this sme process if we rotted the regio bout the x-xis d itegrted log the y-xis Stop! Notice how we used the subdivide d coquer process to pproximte the qutity we wish to determie Tht is we hve subdivided the volume ito pproximtig shells whose volume we kow how to compute We hve the refied this pproximtio by usig fier d fier subdivisios Tkig the limit of this process provides the swer to our questio Idetifyig tht limit with itegrl mkes it possible to compute the volume i questio We hve proved THEOREM (The Shell Method) If V is the volume of the solid of revolutio determied by rottig the cotiuous fuctio f (x) o the itervl [, b] bout the y-xis, the V = p xf(x) dx (8) If V is the volume of the solid of revolutio determied by rottig the cotiuous fuctio f (y) o the itervl [c, d] bout the y-xis, the Z d V = p yf(y) dy (9) c Note: The xis of rottio d the vrible of itegrtio re ot the sme i the shell method, eg, whe rottig roud the y-xis, the itegrtio tkes plce log the x-xis This differs from the disk method where the xis of rottio d xis of itegrtio re the sme Exmples We ll do severl exmples to see how the shell method works d compres with the disk method EXAMPLE 1 Cosider the regio eclosed by the curves y = f (x) =x + x, x =, d the x-xis Rotte the regio bout the y-xis d fid the resultig volume SOLUTION We use the shell method becuse the rottio is bout the y-xis If we used the disk method, we would eed to solve for x i terms of y This is ot esily doe here (d, i fct, would likely be impossible for you) This is oe of the most importt dvtges of the shell method: Iverse fuctios re ot required if the fuctio vrible d xis of itegrtio (ot the xis of rottio) re the sme A sketch of the regio d represettive rectgle ppers i Figure 8 Usig Theorem Z Z V = p xf(x) dx = p x(x + x) dx = p x + x dx x 5 = p 5 + x = p 5 + 8 = 9p 15 y = x + x Figure 8: The regio eclosed by the curves y = f (x) =x + x, x =, d the x-xis Rotte the regio bout the y-xis d fid the resultig volume
mth 11 pplictio: volumes by shells: volume prt iii 19 EXAMPLE 1 Let R be regio eclosed by the curves y = f (x) =x d y = x Rotte R bout the y-xis d fid the resultig volume SOLUTION The two curves meet whe x = x ) x x x = x 1 = ) x =, A sketch of the regio d represettive rectgle ppers i Figure 9 If we used the disk method, we would eed to solve for x i terms of y d we would eed to use two itegrls Usig the shell method, however, we c tret the x height of the cylider s the differece i height betwee the two curves: x So by (Theorem ) Z V = p xf(x) dx = p x x x Z dx = p x x dx = p x 18 = p = p! x 8 YOU TRY IT Try Exmple 1(b) usig the disk method Which method did you fid esier? y = y = x / Figure 9: The regio eclosed by the curves y = f (x) =x, y = x Rotte the regio bout the y-xis d fid the resultig volume EXAMPLE 15 Let R be regio eclosed by the curves y = f (x) =rct x d y = p d the y-xis Rotte R bout the y-xis d fid the resultig volume SOLUTION The two curves meet whe rct x = p ) x = 1 A sketch of the regio d represettive rectgle ppers i Figure Usig shells, the volume is Z V = p xf(x) dx = p x (1 Z rct x) dx = p x x rct xdx=??? We do t kow tiderivtive for x rct x If we use the disk method, we eed to solve for x i terms of y Here y = rct x so x = t y A represettive rectgle is horizotl d is show i Figure By Theorem Z p/ Z p/ V = p [g(y)] dy = p (t y) dy = p sec y 1 dy p/ = p (t y y) = p 1 We used the trig idetity t y = sec y 1 i the work bove This exmple shows tht it is importt to be fmilir with both the shell d disk methods Oe or the other my be most pproprite i give situtio = p p p p/ 1 y = rct x Figure : The regio eclosed by the curves y = f (x) =rct x, y = p d the y-xis Rotte the regio bout the y-xis d fid the resultig volume Which method is most pproprite: shells or disks? The represettive rectgle is for the disk method EXAMPLE 1 Let R be regio eclosed by the curves y = f (x) =e x, x = 1, d the x-xis Rotte R bout the y-xis d fid the resultig volume
mth 11 pplictio: volumes by shells: volume prt iii SOLUTION A sketch of the regio d represettive rectgle ppers i Figure 1 Usig shells, the volume is e y = e x V = p xf(x) dx Z 1 = p xe x dx (let u = x ) Z 1 = p e u du = p (e 1) Notice we eeded to use simple u-substitutio: u = x, du = xdx ) p du = pxdx, d x = ) u = ; x = 1 ) u = 1 YOU TRY IT 7 Try Exmple 1 usig the disk method Is it possible? 1 1 Figure 1: The regio eclosed by the curves y = f (x) =e x, x = 1, d the x-xis Rotte the regio bout the y-xis d fid the resultig volume YOU TRY IT 8 Try settig up the itegrls for Exmple 17 usig the disk method Is it esy? We ed with exmple usig the shell method for rottio bout the x-xis EXAMPLE 17 Let R be regio eclosed by the curves x = 1 (y 1) d the y-xis Rotte R bout the x-xis d fid the resultig volume x = 1 (y 1) SOLUTION The sidewys prbol log with represettive rectgle is show i Figure Usig shells d itegrtio log the y-xis, the volume is Z d Z Z V = p yf(y) dy = p y(1 (y 1) ) dy = p y(y c Z = p y = p y y y ) dy y dy! = p 8 8 = p Figure : The regio eclosed by the curves x = 1 (y 1) d the y-xis Rotte the regio bout the x-xis d fid the resultig volume YOU TRY IT 9 (Usig the Shell Method) Just set up the itegrls for ech of the followig volume problems Simplify the itegrds where possible () R is the regio eclosed by y = x, y = x +, d the y xis i the first qudrt Rotte R bout the y-xis (b) S is the regio eclosed by y = x + x +, y = x, the y xis, d the x xis i the first qudrt Rotte S bout the y-xis (c) T is the regio eclosed by y = x + x +, y = x, d the x xis i the first qudrt Rotte T bout the y-xis (d) S is the regio eclosed by y = 1 x + d y = x i the first qudrt Rotte S bout the y-xis (e) T is the regio eclosed by y = p x, y = x, the y-xis d the x-xis Rotte T bout the y-xis (f ) V is the regio eclosed by y = p x, y = x, d the x-xis Rotte V bout the y-xis b c d e f Figure : The regios for you try it 9 R S R T T 1 1 V
mth 11 pplictio: volumes by shells: volume prt iii 1 YOU p TRY IT (Do by shells) Let R be the regio i the first qudrt eclosed by y = 9 x d the two xes Fid the volume geerted by rottig R bout the y-xis YOU TRY IT 1 (Do by shells) A smll cl bouy is formed by tkig the regio i the first qudrt bouded by the y-xis, the prbol y = x, d the lie y = 5 x d rottig it bout the y-xis (Uits re feet) Fid the volume of this bouy Compre to you try it 1 YOU TRY IT These two problems re rottios bout the y-xis, so either disks or shells re possible However, i ech cse oly oe of these methods is esy Tht s why it is importt to kow both () Let R be the regio i the right hlf-ple eclosed by y = x d y = x x Rotte this regio bout the y-xis d fid the volume Disks or shells: Oly oe method is possible (Aswer: 5p/) (b) Let R be the regio i the first qudrt eclosed by y = l x, y =, d x = e Rotte the regio bout the y-xis d fid the resultig volume (Aswer: p(e + 1)/) (c) Extr Fu Rotte this regio bout the x-xis d fid the resultig volume (Aswer: p(e )) YOU TRY IT (Extr Credit) Try these () Let y = 1 x o [1, ], where > 1 Let R be the regio uder the curve over this itervl Rotte R bout the x-xis Wht vlue of gives volume of p/? (b) Wht hppes to the volume if!? Does the volume get ifiitely lrge? Use limits to swer the questio (c) Isted rotte R bout the y-xis Wht vlue of gives volume of p/? This is prticulrly esy to do by shells! (d) Isted rotte R bout the lie y = 1 If we let = e wht is the resultig volume? YOU TRY IT (Which is it?) Let R be the regio eclosed by y = x d y = x the right hlf-ple where x () Fid the re of R x i (b) Rotte R roud the y-xis d fid the volume geerted Shells or disks? Oly oe method is possible (Aswer: p/)