Suppose I have a random variable $Y$ who's probability density function is given by: $$f_{Y}(y)=a(y^2+y+1)$$
Where $a$ is some normalization constant. The function exists on the range $-3 \leq y \leq 3 $ and is zero everywhere else.
If I define another random variable $X = Y^2$, how would I determine the cumulative distribution function of $X$?
Just evaluate it by integrating the pdf over the relevant domain.
$$\begin{align}F_{\small X}(x)~&=~\mathsf P(Y^2\leq x)\\[2ex]&=~\mathsf P(-\surd x\leq Y\leq \surd x)\\[2ex]&=~\int_{-\surd x}^{\surd x} f_{\small Y}(y)\,\mathrm d y\\[1ex]&~~\vdots\end{align}$$