It's been a while since I looked at calculus, so I'm not sure what mistake I'm making here. I saw this in a text recently:
$$450\int_0^{\infty} e^{-.7t} dt = 450/0.7$$
I understand $e^{-\infty}$ is just zero, since $e$ is simply a number. I also get that $e^0$ is just $1$, as any number to the zero-th power is 1.
So why isn't the integral of $e^{-.7t} = 0 - 1 = -1$, instead of 1/exponent?
$$\int \mathrm{e}^{-0.7 t} \,\mathrm{d}t = \frac{\mathrm{e}^{-0.7 t}}{-0.7} +C \text{.} $$ You've forgotten the denominator. Then $$\left(\lim_{t \rightarrow \infty} \frac{\mathrm{e}^{-0.7 t}}{-0.7}\right) - \frac{\mathrm{e}^{-0.7 \cdot0}}{-0.7} = \frac{0}{-0.7} - \frac{1}{-0.7} = \frac{1}{0.7} \text{.}$$ (Then recall that there is a factor of $450$ in the original expression.)