Do you actually get back the integrand if you differentiate your right hand side with respect to y?
(Besides that, I think your problem is easiest to answer using the Leibniz integral rule.)
Why? When you make a substitution or a change of variables, you have to check how the new variable (i.e. u) depends on the original one (i.e. h). Otherwise, you'd get ridiculous results like the following (let y=2x)
2=\lim_{y\rightarrow 2} y = \lim_{x\rightarrow 2} 2x=4
EDIT: My point is that...