Marginals and conditional of a uniform distribution

Hello, I have been going over this problem for a while. I might have a correct solution but I'm not sure about it. Can anyone verify if it is correct? if is not, can you point me in the right direction. Thanks in advanced.

Suppose that (X,Y) are uniformily distributed over the interval:

I'm unsure about the limits of integration, This I think is the graph

http://g.imagehost.org/0247/graph.jpg

a) Find the marginals of X and Y:

To find I think I have to use:

and obtain

for

and obtain

Again I'm unsure about the limits of integration

For the second part

b)Find both conditional density functions

All I have left to do is integrate and simplify to obtain the conditional density . Can someone verify the limits of integration again? I will not keep posting the details because if the analysis is right I know the rest and if the analysis is wrong , well, I don't have much time. (Worried) Any help will be greatly appreciated.