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Compute integral

  1. May 17, 2009 #1
    2. Relevant equations



    3. The attempt at a solution

    that is [tex]\int_0^{\sqrt{\pi}}\int_x^{\sqrt{\pi}} \sin(y^2) ~dy ~dx[/tex]

    Reverse the order of the integrals (which is possible since the integrand is positive) :
    [tex]0\leq x\leq y\leq \sqrt{\pi} \Rightarrow [/tex] y ranges from 0 to [tex]\sqrt{\pi}[/tex]

    [tex]0\leq x\leq y\leq \sqrt{\pi} \Rightarrow [/tex] x varies from 0 to y.

    So the integral is now :

    [tex]\int_0^{\sqrt{\pi}}\left(\int_0^y \sin(y^2) ~dx\right)~dy[/tex]

    [tex]=\int_0^{\sqrt{\pi}}\left(\sin(y^2)\int_0^y dx\right)~dy[/tex]
     
    Last edited: May 17, 2009
  2. jcsd
  3. May 17, 2009 #2
    Because you changed the order of integration, you don't have to compute the integral of sin(y^2) but rather y*sin(y^2), which can be done by a simple substitution.
     
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