Am I just not seeing what happened to the negative sign? I've written the question below, the part I was confused about was between the first and second step in the second equation.
Example 2 A 10-year annuity provides continuous payments at a rate of $t^2$ at time $t$. The force of interest is $.03t^2$. Write an expression in integral form for the present value of this annuity.
Solution In general, the present value of an $n$-year annuity with continuous payments at a variable force of interest $\delta_t$ is: $$PV =\int_{0}^{n} f(t)e^{-\int_{0}^t \delta_r dr} dt$$ where $f(t)$ is the rate of payment. In this case, we have: $$ \int_{0}^{10} t^2e^{-\int_{0}^t .03r^2 dr} dt = \int_{0}^{10} t^2e^{[.01r^3]_0^t} dt = \int_{0}^{10} t^2e^{.01t^3} dt$$
Any clarification would be much appreciated!
You are correct: the negative sign should still be there.