CT3 Joint Distribution

How to tackle Probability for Joint random variables - P(1<X+Y<2)
For pdf f(x,y)= exp(-(x+y))

I did Google and found in some examples this limit is used in below form

P(0<X+Y<1) = Integration ( limits from 0 to 1-x) and pdf

Can anyone please confirm what is the correct way to obtain this Probability

 

Easiest way is to do
P (X + Y < 2) - P (X + Y < 1) where

P (X + Y < 2) =

Spoiler

\( \int_{x=0}^{2} \int_{y=0}^{2-x} f(x,y) \ dy \ dx\)

P (X + Y < 1) =

Spoiler

\( \int_{x=0}^{1} \int_{y=0}^{1-x} f(x,y) \ dy \ dx\)

 

Thanks Suraj
If only need to obtain P(X+Y>2/3) then

 

P(X+Y>2/3) = 1 - P(X+Y < 2/3)

Then calculate P(X+Y < 2/3) using the same logic as above.

 

Thanks