[tex]\displaystyle\iint_R\sin(384x^2+486y^2)\,\mathrm dA[/tex]
Notice that Given that [tex]R[/tex] is an ellipse, consider a conversion to polar coordinates:
[tex]\begin{cases}x(r,\theta)=\frac r8\cos\theta\\y(r,\theta)=\frac r9\sin\theta\end{cases}[/tex]
The Jacobian for this transformation is
[tex]J=\begin{bmatrix}\frac18\cos\theta&-\frac r8\sin\theta\\\frac19\sin\theta&\frac r9\cos t\end{bmatrix}[/tex]
with determinant [tex]\det J=\frac r{72}[/tex]
Then the integral in polar coordinates is
[tex]\displaystyle\frac1{72}\int_0^{\pi/2}\int_0^1\sin(6r^2\cos^2t+6r^2\sin^2t)r\,\mathrm dr\,\mathrm d\theta=\int_0^{\pi/2}\int_0^1r\sin(6r^2)\,\mathrm dr\,\mathrm d\theta=\boxed{\frac{\pi\sin^23}{864}}[/tex]
where you can evaluate the remaining integral by substituting [tex]s=6r^2[/tex] and [tex]\mathrm ds=12r\,\mathrm dr[/tex].
