Finding Volume of the Solid-washer methodThe region between the graphs of y = x 2 and y = 3 x is rotated around the line x = 3.What is the volume of the resulting solid?I drew the picture, and I saw that I should be using the washer method. Since the region was being flipped at x = 3, I would have to make sure I am subtracting the two functions from 3 and then subtract those two, but I could not seem to get the right answer.

Question

Finding Volume of the Solid-washer methodThe region between the graphs of   y  =      x    2   and   y  =  3  x is rotated around the line   x  =  3.What is the volume of the resulting solid?I drew the picture, and I saw that I should be using the washer method. Since the region was being flipped at   x  =  3, I would have to make sure I am subtracting the two functions from 3 and then subtract those two, but I could not seem to get the right answer.

erlentzed · Accepted Answer

Step 1Using washers, the interval of integration will be along the y-axis from   y  =  0 to   y  =  9. The inner radius as a function of the y-value will be   r  (  y  )  =  3  −      y  , and the outer radius will be   R  (  y  )  =  3  −  y      /    3; thus the differential volume of a washer with thickness dy is  d  V  =  π  (  R  (  y      )    2    −  r  (  y      )    2    )    d  y  =  π  (  (  3  −  y      /    3      )    2    −  (  3  −      y        )    2    )    d  y  ,and the total volume is given by the integral  V  =  π      ∫          y      =      0        9    (  3  −  y      /    3      )    2    −  (  3  −      y        )    2      d  y  .Step 2Using cylindrical shells, the interval of integration will be along the x-axis from   x  =  0 to   x  =  3. The height of the shell is given by the difference of the functions   y  =      x    2   and   y  =  3  x, so   h  (  x  )  =  3  x  −      x    2   (since   3  x  ≥      x    2   on   x  ∈  [  0  ,  3  ]). The circumference of a representative shell is simply   2  π  (  3  −  x  ), thus its differential volume is   d  V  =  2  π  (  3  −  x  )  (  3  x  −      x    2    )    d  x  ,and the total volume is  V  =      ∫          x      =      0        3    2  π  (  3  −  x  )  (  3  x  −      x    2    )    d  x  .Both calculations give the same result.

ajakanvao · Accepted Answer

Explanation:Volume by washers can be used in this case if you stack your washers parallel to the x-axis, centered at   x  =  3. Identify the outer radius R (as a function of y, since every washer is determined by how high it&#039;s sitting in the stack) and the inner radius (again, as a function of y); the volume of each thin washer is   π  (      R    2    −      r    2    )    Δ  y and the total volume is   π      ∫    0    9    (      R    2    −      r    2    )    d  y.

The region between the graphs of y=x^{2} and y=3x is rotated ariund the line x=3.

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