Check my work? Finding line that divides rotational volume into equal parts. Not getting right answer.I need to rotate the area between the curve y = x 2 and y = 9, bounded in the first quadrant, around the vertical line x = 3. I then must find the height (m) of the horizontal line that divides the resulting volume in half.I've been trying to set up two integrals with washers. One from ∫ m 9 3 2 − ( 3 − x ) 2 d y and the other for the bottom region ∫ 0 m 3 2 − ( 3 − x ) 2 d y. The 3 is the outer radius of the washer and the 3 − x gives the inner hole of the washer. I can't seem to get the right answer. π ∫ m 9 ( 6 y − y ) d y = π ( 4 y 3 2 − 1 2 y 2 ) evaluated from 9 to m = π ( 135 2 − ( 4 m 3 2 − 1 2 m 2 ). Similarly, for the bottom region integral, I get π ( 4 m 3 2 − 1 2 m 2 ). I then try to set the volumes equal to each other and solve for m.I believe the answer I should get is 9 2 3 but I don't get that value for m.If I continue, I get 135 2 = 8 m 3 2 - m 2 but I don't see any easy way to solve without using a calculator.

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Check my work? Finding line that divides rotational volume into equal parts. Not getting right answer.I need to rotate the area between the curve   y  =      x    2   and   y  =  9, bounded in the first quadrant, around the vertical line   x  =  3. I then must find the height (m) of the horizontal line that divides the resulting volume in half.I&#039;ve been trying to set up two integrals with washers. One from       ∫          m              9            3    2    −  (  3  −  x      )    2    d  y and the other for the bottom region       ∫          0              m            3    2    −  (  3  −  x      )    2    d  y. The 3 is the outer radius of the washer and the   3  −  x gives the inner hole of the washer. I can&#039;t seem to get the right answer.  π      ∫          m              9        (  6      y    −  y  )  d  y  =  π  (  4      y          3      2        −      1    2        y    2    ) evaluated from 9 to m   =  π  (      135    2    −  (  4      m          3      2        −      1    2        m    2    ). Similarly, for the bottom region integral, I get   π  (  4      m          3      2        −      1    2        m    2    ). I then try to set the volumes equal to each other and solve for m.I believe the answer I should get is       9          2      3       but I don&#039;t get that value for m.If I continue, I get       135    2   =  8      m          3      2       -       m    2   but I don&#039;t see any easy way to solve without using a calculator.

sellk9o · Accepted Answer

Step 1The volume of a rotational solid is   π  ∫      r    2        d    x, where x is an arbitrary variable and r is the radius in terms of x. Now, somewhat counterintuitively, let&#039;s put r in terms of y. For any value of   y  ∈            R        +  ,   x  =      y  . The radius of the solid is the distance between   x  =  3 and   f  (  y  )  =      y  , which is given by   3  −      y  .Step 2Thus, we have                     V                            =        π                  ∫          0          9                (        3        −                  y                          )          2                          d                y                                          =        π                  (                      ∫            0            9                    (          9          +          y          )                      d                    y          −                      ∫            0            9                    6                      y                                d                    y          )                                                  =        π                  (                                                  9              y              +                              1                2                                            y                2                            |                        0            9                    −                                                  4                              y                                  3                                      /                                    2                                            |                        0            9                    )                                                  =        π                  (          81          +          40                      1            2                    −          108          )                =        13.5        π            The reason you&#039;re off is, as far as I can tell, you&#039;re still rotating around   x  =  0 instead of rotating about   x  =  3, as the question dictates

Rotate the area between the curve y=x^2 and y=9, bounded in the first quadrant, around the vertical line x=3. Then find the height (m) of the horizontal line that divides the resulting volume in half.

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