Chapter 14.7, Problem 26ES

The transformation $x=au,y=b\upsilon \left(a>0,b>0\right)$ can be rewritten as $x/a=u,y/b=\upsilon ,$ and hence it maps the circular region $u^2+{\upsilon }^2\le 1$ into the elliptical region $\frac{x^2}{a^2}+\frac{y^2}{b^2}\le$ In these exercises perform the integration by transforming the elliptical region of integration into a circular region of integration and then evaluating the transformed integral in polar coordinates.

${\displaystyle \underset{R}{\iint }{e}^{-\left(x^2+4y^2\right)}dA},$ where R is the region enclosed by the ellipse $\left(x^2/4\right)+y^2=1.$