Finding $P(X+Y \geq 0.5)$ given $f(x,y) = x - y + 1$ for $0 \leq x, y \leq 1$

Question

For the bivariate density function (not necessarily independent) $$f(x,y) = x - y + 1$$ for $$0 \leq x, y \leq 1$$

I am trying to find $\Pr(X+Y \geq 0.5)$.

I integrated $y$ across $0.5-x$ to $1$ and then integrating that from $0$ to $1$ $$\int^{1}_{0}\int^{1}_{0.5-x}(x-y+1) \,dy\,dx$$

But this seems to be wrong when I go through the calculation. I don't know what I could have done wrong here?

Answer 1

Does this plot of the region help?