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

  • Thread starter truman
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  • #1
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Homework Statement


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

Homework Equations


Please see attachment.


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.

Homework Statement





Homework Equations





The Attempt at a Solution

 

Attachments

Answers and Replies

  • #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].
 
  • #3
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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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