Joint density. Help!

[Actuarial_Deepika]

Let x and y have the joint density:
f(x,y) = 6/7(x+y)^2 for 0<=x<=1 and 0<=y<=1

a.Find the marginal densities of X and Y. .
b.By integrating over the appropriate regions, find:
i)P(X>Y)
ii)P(X+Y)<=1
iii)P(x>=1/2)


For this one, I got the answer for part(i). That is 1-P(X<=Y).
but even for that, why do we take limits for the double integration as 0 to 1 and then x to 2?

 

[BGM]

Actually you just direct translate the inequality..
If you wish, you can draw the region in the unit square as well.

i) \( \Pr\{X > Y\} = \int_0^1 \int_y^1 f_{X,Y}(x,y)dxdy
= \int_0^1 \int_0^x f_{X,Y}(x,y)dydx \)

ii) \( \Pr\{X+Y \leq 1\} = \int_0^1 \int_0^{1-y} f_{X,Y}(x,y) dxdy
= \int_0^1 \int_0^{1-x} f_{X,Y}(x,y) dydx \)

iii) \( \Pr\left\{X \geq \frac {1} {2} \right\}
= \int_{\frac {1} {2}}^1 \int_0^1 f_{X,Y}(x,y) dydx
= \int_{\frac {1} {2}}^1 f_X(x)dx \)