jmc

Since the following inequalities are true, $$\sqrt{1}<\sqrt{2}<\sqrt{4}\Rightarrow 1<\sqrt{2}<2$$$$\sqrt{16}<\sqrt{22}<\sqrt{25}\Rightarrow 4<\sqrt{22}<5$$$$\sqrt{196}<\sqrt{222}<\sqrt{225}\Rightarrow 14<\sqrt{222}<15$$the smallest integer greater than $\sqrt{2}$ is $2$, the smallest integer greater than $\sqrt{22}$ is $5$, and the smallest integer greater than $\sqrt{222}$ is $15$. Therefore, $2+5+15=\boxed{22}$.