I came across the following formula and I have had trouble working it myself:
$$ \mathbb{E}[f(x + \delta u)u] = \frac{\int_{\delta \mathbb{S}} f(x + u) \frac{u}{\lvert\lvert u \rvert\rvert}\ du}{Vol_{d-1}(\delta \mathbb{S})}$$
where $d$ is the dimension of $f$, $S$ is the unit sphere, $\delta > 0$ and $u$ are unit vectors.