If we get to use calculus, I'd evaluate this:
\(\displaystyle \int_{-\infty}^{\infty} y\cdot f_{Y}(y)\;dy\)
Of course, I would also check for validity by verifying:
\(\displaystyle \int_{-\infty}^{\infty} f_{Y}(y)\;dy\) = 1
Without the calculus, the geometry is not difficult.
\(\displaystyle \frac{1}{2}\cdot\frac{3}{4} + \frac{5}{2}\cdot\frac{1}{4}\)
Unique answers don't care how you find them.