6.2 - Math 182 Homework Section 6.2 Problem 8 Find the...

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Unformatted text preview: Math 182 Homework Section 6.2 Problem 8 Find the volume of the solid formed by taking the region enclosed by y = x 2 3 , x = 1 and y = 0 and rotating it around the y-axis. Notice that since 1 2 3 = 1 and 0 2 3 = 0 the boundaries of the region are at (0 , 0) and (1 , 1). Notice that each vertical rectangle rotates to produce a shell. This shell has a thickness of dx , a height which is the difference of y coordinates y constant 1- y 2 3 function = 1- x 2 3 . The radius of this circular shell is just x and hence we get that the volume of the shell at position x is 2 πrh · thickness = 2 πx (1- x 2 3 ) dx = 2 π ( x- x 5 3 ) dx Since there are shells for each x in the range from 0 ≤ x ≤ 1 the volume is given by 1 = 2 π integraldisplay 1 parenleftBig x- x 5 3 parenrightBig dx = 2 π bracketleftBigg x 2 2- 3 8 x 8 3 bracketrightBiggvextendsingle vextendsingle vextendsingle vextendsingle vextendsingle x =1 x =0 = 2 π parenleftbigg 1 2- 3 8- parenrightbigg = 2 π 4- 3 8 = π 4 Problem 10...
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This note was uploaded on 09/21/2009 for the course MATH 182 taught by Professor Keppelmann during the Spring '08 term at Nevada.

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6.2 - Math 182 Homework Section 6.2 Problem 8 Find the...

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