There is a part in my notes where (when finding the distributional Laplacian of $1/|\textbf{x}|$) we note that $$\int_{B_\epsilon (0)} \phi \, \mathrm{d}x \approx 4\pi\epsilon^2 \phi(0)$$ and $$\int_{B_\epsilon (0)} \frac{\partial \phi}{\partial n}\, \mathrm{d}x =\mathcal{O}(\epsilon^2).$$ Supposedly this is true for small $\epsilon$ because $B_\epsilon (0)$ is the surface of the sphere.
I don't understand either of these integrals. Please help.
For the first one, observe \begin{align} \phi(x) = \phi(0) + \nabla\phi(0)\cdot x+\mathcal{O}(|x|^2) \end{align} then \begin{align} \int_{B_\varepsilon(0)}\phi(x)\ dx \approx& \int_{B_\varepsilon(0)} \phi(0)\ dx +\int_{B_\varepsilon(0)}\nabla\phi(0)\cdot x\ dx +\text{ remainder}\\ \approx&\ \phi(0)\operatorname{Area}(B_\varepsilon(0)) = 4\pi\varepsilon^2 \phi(0) \end{align} since \begin{align} \left|\int_{B_\varepsilon(0)} \nabla\phi(0)\cdot x\ dx\right| \leq C\int_{B_\varepsilon(0)} |x|\ dx \leq C\varepsilon^3 \end{align} and likewise for the remainder term.
The second equation follows the exact same line of argument.