(Solved): Problem 4. One of the important functions in physics and mathematics is the Gaussian function: g(x) ...

Problem 4. One of the important functions in physics and mathematics is the Gaussian function: $g(x)=e^{?x^2}.$ There is no elementary antiderivative function for $g(x)$. Therefore, we cannot calculate the following integr: directly: $I={?}_0^{?}{e}^{?x^2}dx$ However, the double integral enables us to calculate it as follows: a) We can write $I^2$ as follows $I^2=\left({?}_0^{?}{e}^{?x^2}dx\right)\left({?}_0^{?}{e}^{?y^2}dy\right)={?}_0^{?}{?}_0^{?}{e}^{?x^2?y^2}dxdy.$ Show the relation $\frac{?}{4}\left(1?e^{?a^2}\right)?{?}_0^a{?}_0^a{e}^{?x^2?y^2}dxdy?\frac{?}{4}\left(1?e^{?2a^2}\right)$ for any $a>0$. b) Use the squeeze theorem and conclude $I=\frac{\sqrt{?}}{2}$ when $a??$.