Probability that sum of the squares of two normal variables is less than 1

Question

$X$ and $Y$ are independent standard normal variables. Find $P(X^2+Y^2<1)$.

I know that $X^2+Y^2$ follows a chi-square distribution with $2$ degrees of freedom and am able to write down the pdf as $\dfrac{\exp(-x/2)}{2}$ for $x>0$.

Does integrating the pdf from $0$ to $1$ gives the answer?

If not, how can I compute $P(X^2+Y^2<1)$?