bivariate marginal density function

find the marginal density function for $X$ and $Y$ .

$f(x,y) = \left\{ \begin{array}{rcl} x+y & \mbox{for} & 0 \leq x \leq 1, 0 \leq y \leq 1 \\ 0 & \mbox{for} & \mbox{other} \end{array}\right.$

so far I have

$f(x) = \left\{ \begin{array}{rcl} \int_{0}^{1} x+y \ dy & \mbox{for} & 0 \leq x \leq 1 \\ 0 & \mbox{for} & \mbox{other} \end{array}\right.$

= $f(x) = \left\{ \begin{array}{rcl} xy+\frac{y^2}{2} \ \bigg{|}^{1}_{0} & \mbox{for} & 0 \leq x \leq 1 \\ 0 & \mbox{for} & \mbox{other} \end{array}\right.$ = $f(x) = \left\{ \begin{array}{rcl} x+\frac{1}{2} & \mbox{for} & 0 \leq x \leq 1 \\ 0 & \mbox{for} & \mbox{other} \end{array}\right.$

is this correct?

And how would you find:
$P \left( X \geq \frac{1}{2} \bigg{|}Y \geq \frac{1}{2} \right)$ ?