Checking limits when changing the order of integration

[elle]

Hi, I've got a few integration problems that I would like some help with:

The one I'm currently working on is this:

http://i26.photobucket.com/albums/c109/mathsnerd/4c37af04.jpg"

I'm supposed to change the order of integration but before I start integrating, I want to make sure I've changed the limits correctly. Can someone please check for me? Thank you!

 

[benorin]

The bounds on the first integral read:

$$0\leq x \leq 4 \mbox{ and } \sqrt{x}\leq y \leq 2$$

notice that y is bound below by $\sqrt{x}$

and so the integral you want has bounds:

$$0\leq y \leq 2 \mbox{ and } 0\leq x \leq y^2$$

 

[benorin]

To be clear, the domain of integration is the smaller roughly triangular region in the upper left corner of the box [0,4]X[0,2].

 

[benorin]

Afer changing the bounds, try applying the substitution

$$u=x+y^2\Rightarrow du=dx \mbox{ so that }0\leq x \leq y^2 \Rightarrow y^2\leq u\leq 2y^2$$