I could use some help interpreting the result for $$ \int_{0}^\infty y^2 e^{ay} dy $$
which my textbook says = $-2 / a^3$.
I can see how partial integration leads to $$ \left[\frac{e^{ay}}{a}(y^2 - \frac{2y}{a} + \frac{2}{a^2})\right]\rvert_{y=0}^\infty$$
and then I suppose this leads to $$ (\infty -\infty + \infty) - (0 - 0 + \frac{2}{a^3}) = -\frac{2}{a^3} $$?
The result in the textbook matches a calculation with a<0 in matlab, but I don't really see how:
- a negative result can represent an area under a curve that seems positive everywhere to the right of 0?
- such a small result can represent this area to $\infty$, even when positive, from an exponential curve?