Volume of a solid rotated around axisThe task is finding the volume of the solid found by rotating x = 2 y − y 2 From x = 0 around the axis y = − 1

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Volume of a solid rotated around axisThe task is finding the volume of the solid found by rotating  x  =  2  y  −      y    2  From   x  =  0 around the axis   y  =  −  1

Ezequiel Davidson · Accepted Answer

Step 1We rotate the plane region  A  :=  {  (  x  ,  y  )  :  y  ∈  [  0  ,  2  ]  ,  0  ≤  x  ≤  y  (  2  −  y  )  }around the line   y  =  −  1. By symmetry the centre of mass of A is along the line   y  =  1 and therefore its distance from the axis   y  =  −  1 is   r  =  1  −  (  −  1  )  =  2.Step 2Hence the volume is   V  =  2  π  ⋅  r  ⋅      |    A      |    =            16      π        3   where |A| is the area of A, that is       |    A      |    =      ∫          y      =      0        2    (  2  y  −      y    2    )  d  y  =            [              y        2            −                        y          3                3            ]        0    2    =      4    3    ..

The task is finding the volume of the solid found by rotating x=2y-y^2.

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