How Can I Modify My Equation to Become a Continuous-Time Function?

[sooyewguan]

Lets say I have an equation,

$$y=\alpha e^{\beta W}$$

where,
$$\alpha = a e^{b f}$$ and $$\beta = c f + d$$

$$W = \int^{T}_{0}f dt$$

My problem now is, what happen if $$f$$ is changing with time $$t$$, $$f(t)$$

How do I modify my main equation, $$y$$, so that it become an continuous-time function, $$y(t)$$.

Thank you.

 

[HallsofIvy]

I'm not sure what you mean: what you give
[tex]y= \alpha e^\beta W(t)[/itex]
is a "continuous-time function"- or at least a continuous function of t.

If you want to you can replace each of $\alpha$, $\beta$, and W with their explicit dependence on t:
$$y(t)= ae^{bf(t)} e^{(cf+d)\int_0^t f(u)du}$$
(I've changed the dummy variable in the integral to u so as not to confuse it with the variable t.)

But I don't think that really adds anything as long as you don't know the explicit form of f.