Suppose that you irradiate a beam of monoenergetic alpha particles on a target. You can determine the saturation activity using

Saturation activity = $\frac{R\_a x \sigma \rho N\_A}{A}$

But how do you determine the activity after say half-an-hour of irradiation?

What is confusing me about this is that a radioactive isotope will start to decay as soon as it is produced so I don't know how to quantify this.