Chapter 14.7, Problem 27ES

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.

${\displaystyle \underset{R}{\iint }\sin \left(4x^2+9y^2\right)dA},$ where R is the region in the first quadrant enclosed by the ellipse $4x^2+9y^2=1$ and the coordinate axes.