Finding volume of solid using shell method.Rotating the region bounded by y = x 3 , y = 0 , x = 2 around the line y = 8.I just want to double check if my initial formula is on the right track. V y = ∫ 0 8 ( 8 − y ) ( y 1 3 ) d y

Question

Finding volume of solid using shell method.Rotating the region bounded by   y  =      x    3    ,  y  =  0  ,  x  =  2 around the line   y  =  8.I just want to double check if my initial formula is on the right track.      V    y    =      ∫    0    8    (  8  −  y  )  (      y                  1        3              )  d  y

Ali Harper · Accepted Answer

Explanation:You are on the right track, but I think you want  V  =      ∫    0    8    2  π  r  (  y  )  h  (  y  )  d  y  =      ∫    0    8    2  π  (  8  −  y  )  (  2  −      y          1              /            3        )  d  y

termegolz6 · Accepted Answer

Step 1Since you are rotating around the line   y  =  8, which is parallel to the x-axis, it makes sense to integrate along the x-axis (i.e., integrate dx) rather than along y. The outer radius at the point x (let&#039;s call it R(x)) will be from   y  =  8 to   y  =  0 and the inner radius at the point x (let&#039;s call it r(x)) will be from   y  =  8 to   y  =  0 and the inner radius at the point x (let&#039;s call it r(x)) will be from   y  =  8 to   y  =      x    3  ; this all happens over the region where x goes from 0 to 2.Step 2      ∫    0    2    π  [  R  (  x      )    2    −  r  (  x      )    2    ]    d  x  =      ∫    0    2    π  [  (  8  −  0      )    2    −  (  8  −      x    3        )    2    ]    d  x  =  ⋯

Rotating the region bounded by y=x^3, y=0, x=2 around the line y=8.

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