Two numbers $x,y$ are randomly chosen from the segment [0,5] find the probability that $x+y\ge 5$ and $\left|x-y\right|\le 1$ Two numbers $x,y$ are randomly chosen from the segment [0,5] find the probability that $x+y\ge 5$ and $\left|x-y\right|\le 1$.

So the total area is 25 and the yellow area that has the needed requirements is given by: 
$\int _2^3\:\left(x+1-\left(5-x\right)\right)dx+\int _3^4\:\left(x+1-\left(x-1\right)\right)dx+\int _4^5\:\left(5-\left(x-1\right)\right)dx=\:\frac{9}{2}$
So we devide them and get the probability, anyway is the way I calculated the yellow area correct?

You are correct. The area is easy to get with geometry: the area of the rectangle is $2\sqrt2\times\sqrt2=4$ and that of the triangle is $1\times1/2=1/2$ for a total of $9/2$. Divide by the total possible area ($5^2=25$) and you get a probability
$$
p=\frac{9/2}{25}=9/50=0,18.
$$

