Volume of Solid of Revolution about an equationI learned about the disk and shell method for finding a volume of a solid of revolution. For my question I don't think I can use either method directly. Here is my question:Find the volume of the region bounded by y = x and y = x 2 , but rotated about the equation of y = x.Here the axis of revolution is not a vertical or horizontal line, but rather the equation y = x. One idea I had was to convert this diagonal line into a horizontal or vertical one, but I can't see to do this. Maybe polar coordinates might be useful, but I'm not sure.

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Volume of Solid of Revolution about an equationI learned about the disk and shell method for finding a volume of a solid of revolution. For my question I don&#039;t think I can use either method directly. Here is my question:Find the volume of the region bounded by   y  =  x and   y  =      x    2  , but rotated about the equation of   y  =  x.Here the axis of revolution is not a vertical or horizontal line, but rather the equation   y  =  x. One idea I had was to convert this diagonal line into a horizontal or vertical one, but I can&#039;t see to do this. Maybe polar coordinates might be useful, but I&#039;m not sure.

emarisidie6 · Accepted Answer

Step 1Another approach using a &#039;&#039;modified&#039;&#039; disk method.The distance of a point   (  x  ,  y  )  =  (  x  ,      x    2    ) from the line   y  =  x is:  r  =                    |            x      −              x        2                    |                    2      so the area of a circle orthogonal to the axis of rotation is  A  =  π      r    2    =  π            (      x      −              x        2                    )        2              2  Step 2And the volume of a disk of length   δ  x  =      2    d  x along the axis of rotation is  d  V  =      π    2        ∫    0    1    (  x  −      x    2        )    2    δ  x  =      π          2            ∫    0    1    (  x  −      x    2        )    2    d  xAnd the volume of the solid of revolution is the integral:  V  =      π          2            ∫    0    1    (  x  −      x    2        )    2    d  x

rialsv · Accepted Answer

Step 1Start by writing the curve in parametric form, so   x  =  t  ,  y  =      t    2  . Now use a matrix to rotate the curve by 45 degrees clockwise so that the new curve is also given parametrically.                    (                    x      y                      )              =      1          2            (                            1                          1                                      −          1                          1                      )                      (                    t              t        2                            )            Step 2Now consider the volume as  π      ∫          t      =      0        1        y    2              d      x              d      t        d  t

Find the volume of the region bounded by y=x and y=x^2, but rotated about the equation of y=x.

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