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.