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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!

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You are correct: the negative sign should still be there.

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  • $\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review $\endgroup$ – Yanior Weg Feb 21 at 14:40
  • $\begingroup$ Thanks for that feedback! The question was "Am I just not seeing what happened to the negative sign?", so the OP seemed to basically be asking why is the negative sign gone (they appear to know it should not be gone). I was just confirming this for them, not sure what more there is to say. However, if it is preferred my answer is deleted, I will do so. $\endgroup$ – Minus One-Twelfth Feb 21 at 19:13
  • $\begingroup$ Sorry for this misunderstanding. When I have done this review, I mistook this for a wrongly placed comment (because it looked like one). However, now I see, that your answer is actually OK, as the question was posed in such a way, that your answer is the only possible one. $\endgroup$ – Yanior Weg Feb 21 at 21:59

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