I'm unsure how to calculate a continuously varying, continuously paid annuity. I'll write up my solution (which I suspect is wrong) to one, sample question, and I would greatly appreciate any correction.
The question (problem 29.13 on page 273):
Payments are made to an account at a continuous rate of $(8k+tk)$, where $0\le t\le10$. Interest is credited at a force of interest $\delta_t=\frac1{8+t}$. After 10 years, the account is worth 20,000. Calculate $k$.
My attempt at a solution:
The accumulation function is $e^{\int_0^t(8+s)^{-1}ds}$, which comes to $e^{\left.\ln\left|s+8\right|\right|_0^t}=\frac{t+8}8$, so that the discount (inverse-accumulation) function (to evaluate the present ($t=0$) value) is $\frac8{t+8}$.
Then the present value is the limit of sums of $(\textrm{discount})\times(\textrm{payment})$ i.e. $\int_0^{10}(8k+tk)\frac8{t+8}dt=80k$.
The accumulated value is then $80k\times(\textrm{accumulation(10)})=80k\frac{18}8$; since that's given as $20000$, we have $k=20000/180\approx111.11$.
Could anyone post the right solution or explain what (if anything) is wrong with mine, please?
Another way to look at it is that the accumulated value should be $$20,000=\int_0^{10} (8k+tk) \exp\left(\int_t^{10} \frac{ds}{8+s}\right) dt.$$ This way you can see the payment made at time $t$ accrue interest from time $t$ until time 10.