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`y = x^3, y = 8, x = 0` Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

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The volume of the solid obtained by rotating about x-axis by using cylindrical shell method is

`V = int_a^b 2piyf(y) dy`

The given information is

The curves

 

`y = x^3 =gt x = y^(1/3) `

`y = 8` , `x = y = 0` and rotation is about y-axis

 

`therefore V = int_0^8 2piy[y^(1/3) - 0] dy `

          =`2pi int_0^8 y^(4/3) dy ` 

          = `2pi 3/7 y^(7/3)|_0^8 `

         =  `(6pi)/7 * 8^(7/3)`

         =  `(6pi)/7 * 2^7`

        = `(768pi)/7`

`therefore` The required volume is  `(768pi)/7`

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