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Integration by parts, help me understand why the integration limits changed.

  1. Oct 23, 2011 #1
    1. The problem statement, all variables and given/known data
    I am doing self-study. I am on problem 5.6 #27 in the Stewart text 3rd E.

    I don't understand why the integration limits changed after the given substitution.

    The given substitution was:

    x=θ^2 dx=2θdθ

    2. Relevant equations
    Please see attachment.


    3. The attempt at a solution

    I understand the substitution, and how theta sq became 2theta d theta. What I don't understand is, why when the x was substituted, the integration limits changed from a square root to no square root.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     

    Attached Files:

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  3. Oct 23, 2011 #2

    HallsofIvy

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    The original limits of integration are from [itex]\theta= \sqrt{\pi/2}[/itex] to [itex]\theta= \sqrt{\pi}[/itex]. With [itex]x= \theta^2[/itex] they become [itex]x= (\sqrt{\pi/2})^2= \pi/2[/itex] and [itex]x= (\sqrt{\pi})^2= \pi[/itex].
     
  4. Oct 23, 2011 #3
    Yes, I see. You have to substitute in the variable and then it's squared, removing the square root sign.

    Brain fart. Thank you!
     
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