Limits of Integration Re: Joint Probability Functions

[insixyears]

Edit: I'm not sure if this post was deleted by an admin or if I just didn't click on the submit button. Apologies in advance if it was the former, but I reposted since I couldn't think of a reason why this post would be deleted

Homework Statement

f(x,y)=24xy for 0<= x<=1, 0<=y<=1, 0<=x+y<=1, otherwise x=0

The goal is to show that f(x,y) is a joint probability fxn and I believe we do that by doubly integrating f(x,y) and showing the result equals 1.

The Attempt at a Solution

I believe that the answer is: $$\int_0^1\int_0^{1-y} \! 24xy \, \mathrm{d}x{d}y$$, however I'm having trouble understanding the limits of integration. Why is it that we integrate from 0 to 1-y on x and from 0 to 1 on y?

Here's a similar problem I had earlier:

Homework Statement

f(x,y)=2 0<x<y, 0<y<1
The problem asks the student to determine whether X,Y are independent.

The Attempt at a Solution

So, I set out to find the marginal distribution of x:

$$\int_x^1 \! 2 \, \mathrm{d}y$$ Why do we integrated from x to 1
$$\int_0^y \! 2 \, \mathrm{d}x$$ Why do we integrate from 0 to y

Could we switch the limits of integration as long as we adjust them for both X and Y? For example, could we integrate from 0 to 1 instead of x to 1 if we change the limits of integration on Y?

 

[lippyka]

I'm not entirely sure if this is the right place to ask this question, but I figured it was worth a shot. Thank you in advance for any help!