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Homework Help: Integral and its largest value

  1. Dec 3, 2005 #1
    hello..could you please help me to solve this problem?

    Along what curve of the family y=x^n does the integral
    int{(25xy-8y^2)dx} attain its largest value? and the boundaries for the integral is from (0,0) to (1,1)

    thank you..
     
  2. jcsd
  3. Dec 3, 2005 #2

    Galileo

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    Any thoughts on this problem? Maybe just evaluating the integral along the curve x^n?
     
  4. Dec 3, 2005 #3

    HallsofIvy

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    \ doesn't mean a whole lot to me. Do you mean "where does x^n intersect the integral of 26xy- 8y^2 dx with the largest y value?
     
  5. Dec 3, 2005 #4
    Is this what you're looking for ??

    [tex] \begin{gathered}
    y = x^n \Rightarrow 25xy - 8y^2 = 25x^{n + 1} - 8x^{2n} \hfill \\
    \frac{d}
    {{dn}}\left[ {\int\limits_0^1 {\left( {25x^{n + 1} - 8x^{2n} } \right)dx} } \right] = \frac{{16}}{{\left( {2n + 1} \right)^2 }} - \frac{{25}}{{\left( {n + 2} \right)^2 }} = 0 \Rightarrow \frac{4}{{2n + 1}} = \frac{5}{{n + 2}} \Rightarrow n = \frac{1}
    {2} \hfill \\
    \therefore {\text{Curve is }}y = \sqrt x \hfill \\
    \end{gathered} [/tex]

    (If you allow 'n' to be a rational number, that is :wink:)

    ---?Though I'm not sure this is what you're looking for :frown: ?
     
    Last edited: Dec 3, 2005
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