In the STEP III Statistics module, when the distribution of $Y$ is being worked out given $Y=X^2$, I'm having some trouble understanding the following lines:
$$
\begin{aligned}
\mathrm{F}_Y(y)&=\mathrm{P}(Y \leqslant y)\\
&=\mathrm{P}(X^2 \leqslant y)\\
&=\mathrm{P}(X \leqslant \sqrt{y})\\
&=\int_{-\infty}^{\sqrt{y}} \mathrm{f}(t)\mathrm{d}t
\end{aligned}
$$
I'm confused because I don't see how it isn't as follows:
$$
\begin{aligned}
\mathrm{F}_Y(y)&=\mathrm{P}(Y \leqslant y)\\
&=\mathrm{P}(X^2 \leqslant y)\\
&=\mathrm{P}(-\sqrt{y} \leqslant X \leqslant \sqrt{y})\\
&=\int_{-\sqrt{y}}^{\sqrt{y}} \mathrm{f}(t)\mathrm{d}t
\end{aligned}
$$
I'd be grateful for any clarification you could provide about this situation.
Thanks,
Luke.