Improper Integral

Definitions

$\begin{aligned} \int_{a}^{\infty} f (x) d x & := lim_{t \to \infty} \int_{a}^{t} f (x) d x \\ \int_{- \infty}^{a} f (x) d x & := lim_{t \to - \infty} \int_{t}^{a} f (x) d x \end{aligned}$

If $f (x)$ is continuous at $x \in [a, b]$ except at $x = a$ , $$
\int_{a}^{b} f(x) , dx := \lim_{ t \to a^{+} } \int_{t}^{b} f(x) , dx $$
Same thing goes for if the discontinuity is at $b$
If it is at $c \in (a, b)$ , then $$
\int_{a}^{b} f(x) , dx :=\int_{a}^{c} f(x) , dx +\int_{c}^{b} f(x) , dx

If the limit exists, the integral is convergent. In all other cases, it is divergent
If $f (x)$ is an odd function, $\int_{- \infty}^{\infty} f (x) d x$ may still be divergent
The Volume of Revolution can be convergent, while the integral is divergent??