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Couple Calculus Problems (Mostly Solid Revolutions)

  1. Sep 30, 2008 #1
    1. Find the vol. of a solid generated by revolving the region bounded by [tex]y = 0[/tex] and [tex] = \frac {1}{\sqrt {x}}[/tex] for [tex]1 \le x \le 2[/tex] about the line [tex]y = - 1[/tex]
    2. Find the volume of the solid generated by revolving the region bounded by the lines [tex]x=0[/tex], [tex]y = 0[/tex], [tex]x = 1[/tex], and the curve [tex]y = e^{x^{2}}[/tex]. about the [tex]y[/tex] - axis
    3. Can you write the func. [tex]f(x) = x[/tex] as the product of two differentiable functions ([tex]g(x)[/tex] and [tex]h(x)[/tex]) if [tex]g(0) = h(0) = 0[/tex].
    4. Show that if
    [tex]\frac {a_{0}}{1} + \frac {a_{1}}{2} + ... + \frac {a_{n}}{n + 1}[/tex]
    then
    [tex]a_{0} + a_{1}x + ... + a_{n}x^{n} = 0[/tex]
    for some x in [tex][0,1][/tex]
    5. Find the area bounded by [tex]y = \sqrt {x}[/tex], [tex]\frac {1}{x}[/tex], [tex]\frac {1}{x^{2}}[/tex]
    6. The region in the first quadrant bounded by [tex]x = 2[/tex] and [tex]y = 4[/tex] and [tex]x^{2} = 4y[/tex] is revolved around the [tex]y[/tex]-axis. Find the volume.


    Attempts at solution (so far)
    3. Yes. But I don't know how to show this. It's true for $x^{\frac {1}{2}}$ and $x^{\frac {1}{2}}$ or $x^{\frac {1}{3}}$ and $x^{\frac {2}{3}}$.
     
  2. jcsd
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