Evaluate the double integral by using polar coordinate

Question

Determine the value of $\iint_D \sin(9x^2+4y^2)dA$ where D is the region enclosed by the ellipse $9x^2+4y^2=1$?

How can i evaluate it by changing the region to polar coordinate?

Answer 1

$$\mathrm{ x={r\over3}\cos\phi,\quad y={r\over2}\sin\phi\\ {\partial(x,y)\over\partial(r,\phi)}={r\over6} }$$ The required integral is $$\mathrm{ \int_0^{2\pi}\int_0^1\sin(r^2){r\over6}\,dr\,d\phi\\ ={1\over12}\int_0^{2\pi}\int_0^1\sin(r)\,dr\,d\phi\\ ={\pi\over6}(1-\cos1) }$$