How do you find the volume of the solid obtained by rotating the region bounded by the curves y = 2 x 2 + 5, and y=x+3 and the y-axis, and x=3 rotated around the x axis?

Question

How do you find the volume of the solid obtained by rotating the region bounded by the curves   y  =  2      x          2        +  5, and y=x+3 and the y-axis, and x=3 rotated around the x axis?

Kayley Jackson · Accepted Answer

Sketch the region. Note that   y  =  2      x          2        +  5 is above (greater than) x+3, so the parabola is farther from the axis of rotation.Therefore:At a particular x, the large radius is,   R  =  2      x          2        +  5, and the small radius isr=x+3. The thickness of the disks is dxThe volume of each representative disk would be   π  ∗  r  a  d  i  u      s          2        ∗  t  h  i  k  n  e  s  s. So the large disk has volume:   π  (  2      x          2        +  5      )          2        d  x and the small one has volume   π  (  x  +  3      )          2        d  xThe volume of the washer is the difference, or   π      R          2        d  x  −  π      r          2        d  x and the resulting solid has volume:  V  =  π      ∫          0              3        (  (  2      x          2        +  5      )          2        −  (  x  +  3      )          2        )  d  x  =  π      ∫          0              3        (  4      x          4        +  19      x          2        −  6  x  +  16  )  d  xYou can finish the integral to get             1932      π        5

How do you find the volume of the solid obtained by rotating the region bounded by the curves y

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