jmc

Let the integers be $x$ and $y$, with $x>y$. We have the equations $$\begin{array}{rl}x-y& =6\\ xy& =112\end{array}$$Squaring the first equation, we get $$(x-y{)}^2=6^2\Rightarrow x^2-2xy+y^2=36$$Multiplying the second equation by four, we get $4xy=4\cdot 112=448$. Adding these last two equations, we have $$x^2-2xy+y^2+4xy=36+448\Rightarrow (x+y{)}^2=484\Rightarrow x+y=22$$In the last step, we take the positive square root because both $x$ and $y$ are given to be positive. The sum of the two integers is $\boxed{22}$.