# 6.3 - 6.3: Volumes by Cylindrical Shells (Dated: October 5,...

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6.3: Volumes by Cylindrical Shells (Dated: October 5, 2011) Consider a cylindrical shell with inner radius r 1 , outer ra- dius r 2 , and height h . Its volume is V = π ( r 2 2 - r 2 1 ) h = π ( r 1 + r 2 )( r 2 - r 1 ) h = 2 π r 1 + r 2 2 · h ( r 2 - r 1 ) = 2 π rh Δ r , where r = r 1 + r 2 2 is the average radius and Δ r = r 2 - r 1 is the thinkness of the shell. It follows that one has the volume of the cylindrical shell given by V = circumference × height × thickness. Let S be the solid obtained by rotating about the y -axis the region bounded b the curve y = f ( x ) 0, y = 0, x = a , and x = b , where 0 < a b . The volume of S can be ap- proximated by a Riemann sum whose summands are the volumes of constituent cylindrical shells. To see this, let’s divide the interval [ a , b ] into n subintervals of equal width Δ x = b - a n . Let [ x i - 1 , x i ] be the i th subinterval and let x i be the midpoint of [ x i - 1 , x i ]. Note that if the rectangle with base [ x i -
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## This note was uploaded on 11/11/2011 for the course MATH 152.01 taught by Professor Geline during the Fall '09 term at Ohio State.

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