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

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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 y = 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 · x 2 3 dx = 2 πx 5 3 dx Since there are shells for each x in the range from 0 x 1 the volume is given by 1
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= 2 π integraldisplay 1 0 x 5 3 dx = 2 π 3 8 x 8 3 vextendsingle vextendsingle vextendsingle vextendsingle x =1 x =0 = 2 π 3 8 = 3 π 4 Problem 10 Find the volume of the solid formed by taking the region enclosed by y = 1 x , x = 1 , x = 3 , and y = 0 and rotating it around the horizontal line y = - 1 . Each rectangle rotates to produce a washer - i.e. a disk with a hole in it. The holes all have the
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