Express the integral $$I=\int_{0}^{1} d x \int_{0}^{\sqrt{1-x^{2}}} e^{-x^{2}-y^{2}} d y$$ as an integral in polar coordinates \((r, \theta)\) and so evaluate it.

Transforming integrals from Cartesian coordinates to polar coordinates can make complex integrations simpler. In our given exercise, the original integral is expressed in Cartesian coordinates as \textcolor{blue}{\text{\(I = \text{\textarrow{\text{\textofourtext{\text{\textofourcomponent{\text{\textda{x I = e^{-x^{2}-y^{2}} hyd \theta integral.$$y^-degrees}}}}}}}}}}}} }}}integral}}). To convert this into polar coordinates, we need to make use of the following transformations:

• \)x = r \text{cos} \theta\(
• \)y = r \text{sin} \theta\(

Here, \)r\( represents the radial distance from the origin, while \)\theta\( is the angle measured counterclockwise from the positive x-axis. The region described by \text{0 to \) \text{integral \theta}dx}dx)(0 to is transformed to be positive x-{0 to first \(1}\theta\)}\(}}}}}}}}}})}}})}}))))}}}))}})0}result)} is transformed such that \)0 <= r <= 1\( and \)0 <= \theta <={ \frac{{$\textdict \frac{:]{\theta}$$, from