Expected value of integral with random variable on bound

Question

I'm trying to write the following as a single integral, however I did not get the decired results yet, and tried searching this site without succes. What I'm trying to rewrite is:

$$\mathbb{E}\int^X_0f(y)dy$$ Where X $\in (0, S)$, where $S$ can be both finite or infite and f(y) is a $C^2$ function

So it's really the double integral: $\int^S_0\int^X_0f(y)dyg(x)dx$.

I have tried the following things:

1. Use integration by parts together with the first fundamental theorem of calculus.
2. Rewrite it using the second fundamental theorem of calculus, such that the Expected value ends up in the bound, however this would require Jensen's inequality to hold with equality.
3. Using Tonneli's theorem, however I'm not sure if this would be allowed, since then you can interchange the expected value and the integral, resulting in : $\int^X_0[f(y)G(x)]^S_0dy = \int^X_0f(y)dy$. Which I dont think is correct, since the expected value then disappears.

So my question is how to rewrite this double integral into a single integral, where the integral is w.r.t. y.

I hope someone is willing to help, or to point out which method to look at, all help is appreciated!