Solis of Revolution Solis of revolution re rete tking n re n revolving it roun n is of rottion. There re two methos to etermine the volume of the soli of revolution: the isk metho n the shell metho. Disk metho: In the isk metho, smll retngles tht re perpeniulr to the is of rottion re rotte roun, uiling up series of isks tht stk on eh other in sequene. The with of retngle is epresse s either or. If the re uts the is of rottion, then the height (or length) of retngle (R for rius ) is the istne from the is of rottion to the ounr urve, whih will e the funtion vlue or. If rotting roun the -is, R is funtion of ; if rotting roun the -is, R is funtion of. π [ ] π[ ] ) Fin the volume of the soli forme rotting the region oune +,, n roun the -is. + π[ ] ut it must e in terms of : + π π ( ) π 6π Deprtment of Mthemtis, Sinlir Communit College, Dton, OH
Fin the volume of the soli forme rotting the region oune +,, n roun the -is. + π[ ] ut it must e in terms of : + π ( ) 6π (Note tht the volume n e verifie using the volume formul for sphere: π r /.) ) Fin the volume of the soli forme rotting the region oune,,, n roun the -is. 8 π[ ]. 6 π π ( ) 7 π 7 8π 7 Deprtment of Mthemtis, Sinlir Communit College, Dton, OH
Fin the volume of the soli forme rotting the region oune,,, n roun the -is. 8 L() length of retngle Wht hppens if we rotte this re roun the -is? We still nee to use the length of the retngle euse tht s wht is forming the soli, ut now it s no longer just the istne to the is of rottion euse there s gp the region is not right up ginst the is of rottion. So we swith to moifie isk metho, lle the wsher metho (isk + hole). In the wsher metho, we use the istne from the is of rottion to the outer urve n the istne from the is to the inner urve. r() Wsher: π[ ] [ ] r() is onstnt for this re it s just. r(), ut it must e in terms of / r() Don t forget to use the limits of integrtion for, not! 8 π [ ] / π 8 / π 5 5/ 8 6π 5 Deprtment of Mthemtis, Sinlir Communit College, Dton, OH
) Let s look t the re forme the intersetions of the urves n. If we rotte this re roun the -is, then the upper urve (frthest from the is) is n the lower urve (nerest the is) is. n r(). π[ ] [ ] r() r() 6 π π 7 π 7 5π. If we rotte this re roun the -is, the upper urve (frthest from the is) is n the lower urve (nerest the is) is. The istnes from the is to these urves re -vlues, ut the must e written s funtions of. r() n r(). [ ] [ ] π r() / / π π 5/ 5 π 5 5 π.57 5 Deprtment of Mthemtis, Sinlir Communit College, Dton, OH
) One more: lulte the volume forme revolving the region oune, n out the line. The istne from the is of is just. +,, Ais of rottion The istne r() from the is is ( ) for the funtion r() - + + r() [ ] [ ] [ ] π r() π + π 6 + (+ ) π 6ln(+ ) + π [ ln ¾] +.77 Shell metho: In the shell metho, smll retngles tht re prllel to the is of rottion re rotte roun, uiling up series of shells tht uil the soli from the insie out. The with of retngle is gin epresse s either or. P() or p() is the istne from the retngle to the is of rottion; h() or h() is the length (or height) of the retngle. Gps etween the region n the is re ounte for in the limits of integrtion, so o not require etr lultion s in the wsher metho. The shell metho e to the isk metho gives ou hoie of whihever integrtion woul e esier integrting with respet to or to. Sometimes ou n t (esil) solve for in terms of or vie vers, n re fore into speifi metho. p()h() π π p()h() h() p() h() p() Deprtment of Mthemtis, Sinlir Communit College, Dton, OH 5
) Fin the volume forme revolving the region oune,,, n + out the is. p() h() + π p()h() π + h() () π π ( ) + rtn π p() If ou trie to o this using the isk metho, ou woul first hve to split the re into two prts: retngle up to ½ n the funtion urve from there to. For the ottom prt, π[ ] For the top prt, ½ / π + π tr using some lulus softwre! / ) Fin the volume of the soli proue rotting the region oune,, n roun the line -. Shell metho: p() + h() h() p()h() π π ( + ) π (+ )( ) π + π p() - Deprtment of Mthemtis, Sinlir Communit College, Dton, OH 6
Disk metho: r() + + π[ ] [ ] [ ] r() π + π 8 r() - π 8 / π Neither of these integrtions is ver iffiult, ut the shell metho iels somewht simpler lultions. ) Fin the volume of the torus generte revolving the irle given + roun the line. π p()h() We n work with just the top hlf of the irle n then oule the volume. h() p() Shell metho: p() h() () π ( ) π The first integrl n e otine from tle or reognizing it is simpl the re of the irle πr π. The seon integrl n e solve using u sustitution: π(π) π ( ) / + Deprtment of Mthemtis, Sinlir Communit College, Dton, OH 7
Disk metho: This is lso ole using the isk metho n se on the smmetr of the irle. π[ ] [ ] r() + + r() r() () π + π 8 6π The integrl n gin e otine from tle or reognizing it is the re of qurter irle πr / π/ π (6π) π ) Fin the volume of the soli proue rotting the region oune,,, n roun the -is. For this funtion, nnot e esil solve for in terms of, so the shell metho is require. p() p() sine is negtive in this region, h() - - + + p()h() π π ( + + ) h() π ( + + ) 5 π + + 5 π. Deprtment of Mthemtis, Sinlir Communit College, Dton, OH 8