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

[truman]

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.

 

[HallsofIvy]

The original limits of integration are from $\theta= \sqrt{\pi/2}$ to $\theta= \sqrt{\pi}$. With $x= \theta^2$ they become $x= (\sqrt{\pi/2})^2= \pi/2$ and $x= (\sqrt{\pi})^2= \pi$.

 

[truman]

Yes, I see. You have to substitute in the variable and then it's squared, removing the square root sign.

Brain fart. Thank you!