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**ragnar** So I've done a few change of bases and they're hard but I'm getting the hang of them. Now I have this: $\displaystyle \displaystyle \int \int_{R} \sin(9x^{2} + 4y^{2}) \,\, dA$ where $\displaystyle R$ is the region in the first quadrant bounded by the ellipse $\displaystyle 9x^{2} + 4y^{2}$. It can't be any coincidence that the function I'm integrating is almost verbatim the equation of the region I'm integrating over. Still, in all previous examples I've been able to find out what region I've been applying my transformation to by tracing outlines. I can't do that here. Unless I'm supposed to do this by trig functions... Which is a bit daunting. $\displaystyle x = \frac{1}{3}\cos \theta, y = \frac{1}{2} \sin \theta$? But then how do I do traces? Set $\displaystyle r = 0$ and get just the point, then set $\displaystyle r = x^{2} + y^{2}$? This all seems like shaky ground to me.