Lecture Notes 9 - 6.2 Volumes by Cylindrical Shells Summing the volumes of Riemann Sum 21 3 of the individual cylindrical shells over the interval[0 3

6.2 Volumes by Cylindrical Shells ∆g G ± ²³´²µ¶·¸´¸¹²¸ º »¸³¼»½ º ½»³²¾¹¸¿¿ ± 2ÀÁ1 Â Ã G Ä Å Á3Ã G Æ Ã G Ç Ä Å ∆Ã Summing the volumes of ∆g G of the individual cylindrical shells over the interval [0, 3] gives the Riemann Sum

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g ∆G ± ² g 2³´µ ± ¶ 1·´3µ ± ¸ µ ± ¹ ·∆µ º ±»¼ º ±»¼ Taking the limit as the thickness ∆µ ½ 0 gives the volume integral G ² lim º½¾ g 2³´µ ± ¶ 1·´3µ ± ¸ µ ± ¹ ·∆µ º ±»¼ ² ¿ 2³´µ ¶ 1·´3µ ¸ µ À ·Áµ À Â ² ¿2³´3µ ¹ ¶ 3µ ¸ µ À ¸ µ ¹ ·Áµ À Â ² 2³ ¿´2µ ¹ À ·Áµ À Â ² 2³ Ã 2 3 µ À ¶ 3 2 µ ¹ ¸ 1 4 µ Ä Å 3 0 ² 45³ 2

∆g G ± 2² ³ ´µ¶·´¸¶ ¹º¶»» ·´¼½¾¹ ³ ¹º¶»» º¶½¸º¿ ³ ¿º½ÀÁÂ¶¹¹ 2² Ã ÄÀ G Å ÆÇ Ã ÈÄÀ G Ç Ã ∆É G We approximate the volume of the solid S by summing the volumes of the shells swept out by the n rectangles based on P: g Ê Ë ∆g G Ì GÍÎ