An amount of money $M$ compounded continuously at interest rate $r$ increases according to the law $M(t)=Ce^{rt}$ ($t$ = time in years, $C$ = the initial amount).
What is the average amount of money A in the bank over from 0 to T years?
I thought I would do something like:
$$A = {C \over T} \int_{0}^{T} e^{rt}$$
Evaluating with respect to $t$ I think? Would give me:
$$Ce^{rT} \over Tr$$
But I'm getting the incorrect answer. How do I go about doing this?
Best Answer
$$ A=\frac{1}{T}\int_0^TC \mathrm e^{rt}\mathrm d t=\frac{C}{T}\left[\frac{\mathrm e^{rt}}{r}\right]_0^T=\frac{C}{T}\left[\frac{\mathrm e^{rT}}{r}-\frac{\mathrm e^{r\cdot 0}}{r}\right]=\frac{C}{Tr}\left(\mathrm e^{rT}-1\right) $$