Continuous probability - calculate probability of r.v and distribution function

This is the question:

$X$ is a continuous random variable whose probability density function
is given by
$$f(x)=\begin{cases} \frac{1}{9}x^2 & \text{if $0\leq x \leq 3$}.\\ 0 & \text{otherwise}. \end{cases}$$

(a) what is the probability that $X$ is less than $1$?
(b) Write down the distribution function $F_{X}(x)$ for $X$ (remember to
include the values for $F_{X}$ for all real $x$).

So for (a), I am looking for
$P\{X < 1\} = \frac{1}{9}\int_{-\infty}^{1}x^2\,dx$
however how do I compute the definite integral with negative infinity? Am I allowed to replace the negative infinity with $0$ since the function ranges from $0$ to $3$?

For (b), I am not sure how do this exactly, well my book says "derive and then differentiate the distribution function".