Let C be the curve of intersection of the parabolic cylinder x 2 = 2 y, and the surface 3 z = x y. Find the exact length of C from the origin to the point (2, 2, 4/3).

Question

Let C be the curve of intersection of the parabolic cylinder       x    2    =  2  y, and the surface   3  z  =  x  y. Find the exact length of C from the origin to the point (2, 2, 4/3).

Jimmy Sandoval · Accepted Answer

x    2    =  2  ycan be written as  y  =            x      2        2  let x=t  y  =            t      2        2    3  z  =  x  y  z  =      3          x      y        z  =            t      ⋅                        t          2                2              3    =            t      3        6    r  (  t  )  =&amp;lt;  x  (  t  )  ,  y  (  t  )  ,  z  (  t  )  &amp;gt;    r  (  t  )  =&amp;lt;  t  ,            t      2        2    ,            t      3        6    &amp;gt;      r    ′    (  t  )  =      d          d      t        &amp;lt;  t  ,            t      2        2    ,            t      3        6    &amp;gt;=&amp;lt;      d          d      t        (  t  )  ,      d          d      t        (            t      3        6    )  &amp;gt;=&amp;lt;  1  ,  t  ,            t      2        2    &amp;gt;      |        r    ′    (  t  )      |    =            1      2        +          t      2        +    (                  t        2            2              )      2              |        r    ′    (  t  )      |    =      (    1    +                  t        2            2              )      2              |        r    ′    (  t  )      |    =  (  1  +            t      2        2    )Origin is at t=0(2,2,4/3) is at t=2Length of the curve  L  =      ∫    a    b        |    r  ‘  (  t  )      |    d  t    L  =      ∫    0    2    (  1  +            t      2        2    )  d  t    L  =  [  t  +            t      3              3      ⋅      2            ]    0    2      L  =  [  2  +            2      3              3      ⋅      2        −  0  ]    L  =  [  2  +      4    3    ]    L  =      10    3

Let C be the curve of intersection of the parabolic cylinder x^2 = 2y, and the surface 3z = xy. Find the exact length of C from the origin to the point (2, 2, 4/3).

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