How do you find the volume of the solid generated by revolving the region bounded by y = 2 x − 1 y = x and x=0 and revolve about the y-axis?

Question

How do you find the volume of the solid generated by revolving the region bounded by   y  =  2  x  −  1  y  =      x   and x=0 and revolve about the y-axis?

Samantha Reid · Accepted Answer

To avoid doing 2 integral (which the disc method would require), I would use cylindrical shells.A representative slice parallel to the y axis has area   (      x    −  (  2  x  −  1  )  )  d  xRevolving around the y axis, gives us a shell of volume:  2  π  x  (      x    −  2  x  +  1  )  d  x. The volume of the solid is:  2  π      ∫          0              1        x  (      x    −  2  x  +  1  )  d  x  =  2  π      ∫          0              1        (      x                  3        2              −  2      x          2        +  x  )  d  xThat&#039;s a straightforward integral to evaluate and finish the problem.You should get             7      π        15  Alternative:You could use discs (washers), but you&#039;l have to       ∫          −      1              0       using thedifference between the lines x=0 and    x  =            y      +      1        2   And then      ∫          0              1       using the difference between   x  =            y      +      1        2   and   x  =      y          2

How do you find the volume of the solid generated

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