First you can easily check that $Y$ is non-negative almost surely by checking its CDF \( G_Y \). Then the expectation of such a random variable can be expressed as

\( E[Y] = \int_0^{\infty} [1 - G_Y(y)]dy \)

Also, the expectation of \( h(X) \) can be expressed as

\( E[h(X)] = \int_0^{\infty} h(x)dF_X(x) \)

With these facts I think you can fill all the details.