The integral ${\displaystyle \underset{-}{\overset{}{}}}{e}^{-x^2}dx$ is very important in the study of probability and statistics, but it cannot be evaluated using elementary methods. You will evaluate it by means of a double integral:

$($a$)$ Calculate the polar integral ${}_D{e}^{-r^2}dA,$ where $D$ is a circle of radius $R$ centered at the origin.

$($b$)$ Take the limit of your answer to part $($a$)$ as $R.$

$($c$)$ Express the integral of $e^{-r^2}$ over the entire $xy-$plane in rectangular coordinates, then write this as a product of two integrals.

$($d$)$ By equating your answers to parts $($b$)$ and $($c$),$ determine the exact value of ${\displaystyle \underset{-}{\overset{}{}}}{e}^{-x^2}dx.$