Generally speaking, approximating a function in a calculus problem with another function which looks similar (in some sense or another) is dangerous, since while the functions may look similar, a lot of their attributes are probably still be very different. For example, if you calculate the 4-th derivative of a function $f\left(x\right)$, in many cases it will be VERY different from the 4-th derivative of $g\left(x\right)$ which may look very similar to $f$.
Now, assume you have an integral of the following form: $$I=\int_{y\in\mathcal{Y}}\delta\left(y-y_{0}\right)\left[\int_{x\in\mathcal{X}}f\left(x,y\right)dx\right]dy\quad ; \quad y_0\in\mathcal{Y}$$ Also assume that you cannot perform the integral over $x$ analytically, but the function $f\left(x,y\right)$ is adequately approximated (for $x\in\mathcal{X}$ and $y\in\mathcal{Y}$) by a function $g\left(x,y\right)$, over which you can integrate easily.
Is it legal (as an approximation) to replace $f\left(x,y\right)$ with $g\left(x,y\right)$ in this case? On what conditions?