Chapter 7.2, Problem 64ES

61-66 Suppose that during a period $t_0\le t\le t_1$ years, a company has a continuous income stream at a rate of I(t) dollars per year at time t and that this income is invested at an annual rate of $r\%,$ compounded continuously. The value (in dollars) of this income stream at the end of the time period $t_0\le t\le t_1,$ called the stream’s future value, can be calculated using

$FV={\displaystyle {\int }_{t_0}^{t_1}I(t)e^{r(t_1-t)}dt}$

The present value (in dollars) of the income stream is given by

$PV={\displaystyle {\int }_{t_0}^{t_1}I(t)e^{-r(t-t_0)}dt}$

The present value is the amount that, if put in the bank at time $t=t_{0}atr\%$ compounded continuously, with no additional deposits, would result in a balance of FV dollars at time $t=t_1.$

That is,

$FV=PV\times e^{r(t_1-t_0)}$

In each exercise, (a) find the future value FV for the give income stream I(t) and time period $t_0\le t\le t_1;$ (b) find the present value PV of the income stream over the time period; and (c) verify that FV and PV satisfy the relationship given above.

$I(t)=20,000-200t^2;r=5\%;2\le t\le 10$