how to determine E(y) for the joint probability density function fxy(x,y)=2/19 over the range 0<x<5 and 0<y and x-1<y<x+1?

I tried determining fy for 0<y<1 (fy1) and for 6>y>1 (fy2) separating the space where fxy is defined by the line y=1 and then

I integrated y*fy1dy from 0 to 6 and then y*fy2dy from 1 to 6 is that a correct method to do it? cause I have:

fy1=(y+1)*(2/19)

fy2=(4/19)

and the result I get is not correct? can anyone see where is my mistake?