# Approximate fraction of two integrals

could you propose a way to simplify or approximate (under some assumptions) $\bar{\eta}$ defined as below?

$$\bar{\eta} = \frac{\int f(t)dt}{\int\frac{f(t)}{\eta{(t)}}dt}$$

The $f(x)$ and $\eta(t)$ are generally unknown functions over time $t$ and $\bar{\eta}$ can be understood as overall $\eta(t)$.

Note that $\bar{\eta}$ is not average of $\eta{(t)}$ unless $\eta(t)$ is constant.

I feel that one would need more structure to the problem in order to find a solution or good approximation, feel free to propose any assumptions you find fitting.

Thank you

I think you really need more information about $\bar\eta(t)$ here. If you make the assumption that $\bar\eta(t)$ is constant (not sure how reasonable that is without context), then of course you have
$\bar\eta=\frac{\int f(t)dt}{\frac{1}{\bar\eta(t)}\int f(t)dt}=\bar\eta(t)$.
The other problem is that since you don't really know anything about $\bar\eta(t)$, it could be zero at some point on the domain, in which case $\bar\eta$ isn't even defined.
• Thank you for your input. To give you some physical meaning, I am trying to estimate overall efficiency of a machine with instantaneous power on output $f(t)$ and instantaneous efficiency $\eta(t)$. Then, numerator is energy on output side and denumerator is energy on input. I cannot assume to know analytical structure of $\eta(t)$ as it can model different things, but its range is in $0<\eta(t) \le1$. Also, it is safe to assume that the function is not always continuous. Finally, I might be able to estimate probability distribution function of $\eta(t)$, if need be. Jul 7, 2015 at 9:15