Volumes by Cylindrical Shells: the Shell Method


 Roy Hines
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1 olumes Clinril Shells: the Shell Metho Another metho of fin the volumes of solis of revolution is the shell metho. It n usull fin volumes tht re otherwise iffiult to evlute using the Dis / Wsher metho. Generl formul: (shell rius (shell height The Shell Metho (out the is The volume of the soli generte revolving out the is the region etween the is n the grph of ontinuous funtion f (, is [ rius] f ( Similrl, The Shell Metho (out the is The volume of the soli generte revolving out the is the region etween the is n the grph of ontinuous funtion f (, is [ rius] f ( Comment: An es w to rememer whih metho to use to fin the volume of soli of revolution is to note tht the Dis / Wsher metho is use if the inepenent vrile of the funtion(s n the is of rottion is the sme (e.g., the re uner f (, revolve out the is; while the Shell metho shoul e use if the inepenent vrile is ifferent from the is of rottion (e.g., the re uner f (, revolve out the is.
2 E. Fin the volume of the soli generte revolving out the is the region oune the prol n the is,. 6 ( 6 ( E. Fin the volume of the soli generte revolving out the is the region oune the prol n the is,. The height of the shell ( right urve left urve is. ( ( + Bonus emple : Fin the ove volume using the Dis metho inste. We will first nee to rewrite the funtion in terms of : ; the limits of integrtion re now from to (sine, s goes from to, goes from to. Hene, ( (
3 Revolving out n is the region etween urves The version of Shell metho, nlogous to the Wsher metho, to fin the volume of soli generte revolving the re etween urves out n is of rottion is: (Aout the is The volume of the soli generte revolving out the is the region etween the grphs of ontinuous funtions F( n f (, F( f (, is [ rius] [ F( f ( ] Notie the shell height is now just the ifferene of the heights of the urves. Similrl, the volume of the soli generte revolving out the is the region etween the grphs of ontinuous funtions F( n f (, F( f (, is (Aout the is [ rius] [ F( f ( ] E. Fin the volume of the soli generte revolving the first qurnt region oune,, n the is, out the is. The upper urve is the prol, n is the lower urve. Their first qurnt intersetion is t (,, therefore, (( ( ( 6 6 6
4 Revolving out line other thn the  or is If the is of rottion is not one of the two oorintees, ut rther n ritrr horizontl or vertil line, the volume n e similrl lulte, with some slight justments. E. Fin the volume of the soli generte revolving the region oune,, n, out the line. The is of rottion,, is line prllel to the is, therefore, the shell metho is to e use. The height of the shell is f(, ; n the rius is (s mesure from the is of rottion: when, r, n when, r. Hene, [ rius] ( ( We oul lso fin this volume using the wsher metho. First the funtion nees to e rewritten in terms of : f( /,. The wsher hs n outer rius of R / (s mesure from the is of rottion to the frther w of two urves; whih is the urve on the left; n the inner rius is r (the istne etween the line to the is of rottion. Hene, ( R( ] [ r( ] ([ / ] / [ (9 6 / + / / 9 / 5 / (
5 E. Fin the volume of the soli generte revolving the region oune,,, n, out the line. The is of rottion,, is line prllel to the is, therefore, the shell metho is to e use. The height of the shell is f(, ; n the rius is (s mesure from the is of rottion: when, r, n when, r. Hene, ( ( Sine the istne etween n point to the line k long the is is k, we hve (Shell Metho, out the line k, i.e., line prllel to the is The volume of the soli generte revolving out the line k the region etween the grphs of ontinuous funtions F( n f (, F( f (,, k not etween n, is [ rius] k [ F( f ( ] Note tht we re integrting with respet to, sine the is of rottion is prllel to the is. Similrl, (Aout the line k, k not etween n [ rius] k [ F( f ( ] The is of rottion is prllel to the is, so the integrtion is one with respet to.
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