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Unformatted text preview: Lecture 9: MATH 138-W12-003 January 23, 2012 I) Volumes by shells, Section 6.3 In the previous lecture (or set of notes) we should how to revolve a quadratic curve about the y-axis we needed to complete the square to describe the curve as x ( y ) . This was doable but you can imagine that if we had a cubic or other function it would get a lot harder, if not impossible. This partly motivates the need for an alternative method than disks and washes. What we establish here is the method of volumes by cylindrical shells . Before we get into the calculus portion we first refresh our memories about how to compute the volume of a thin cylindrical shell. If the shell has a height of h , an external radius of r 2 and an internal radius of r 1 then the volume can be written as the difference between the volume of the two solid regular cylinders, V = V 2- V 1 , = r 2 2 h- r 2 1 h = ( r 2 2- r 2 1 ) h, = ( r 2 + r 1 )( r 2- r 1 ) h, = 2 r 2 + r 1 2 ( r 2- r 1 ) h. If we define the mean and the thickness to be r = 1 2 ( r 2 + r 1 ) and r = r 2- r 1 , respectively, then our formula for the volume of a thin shell can be written as, V = 2 rh r . In words this can be interpreted as, V = [Circumference] * [height] * [thickness] . I would argue that the last formula is perhaps the better to remember because if you understand it than you can derive the mathematical version in this particular case or in general....
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