I feel like I almost have a grasp on regions of integration. I am a bit frustrated that I haven't fully gotten it, but I feel like I'm almost there. In this particular homework problem I have a Cartesian integral that I need to convert into a polar integral.
Using polar coordinates, evaluate the integral $$\iint_{R} \sin(x^2 + y^2)\,dA$$ where $R$ is the region $1 \le x^2 + y^2 \le 81$.
In order to convert these to polar coordinates, I substituted $x^2 + y^2 = r^2$ (taking the long way)
\begin{align} x^2+y^2&=(r\sin(\theta))^2(r\cos(\theta))^2 \\ &=r^2(\sin^2(\theta)+\cos^2(\theta))\\ &=r^2(1) = r^2 \end{align}
So I assumed the conversion would be straight-forward
$$\int{}\int_{1}^{9}\sin(r^2)\,r\,dr\,d\theta$$
The inner integral is easy enough to integrate, it only requires a single u-substitution, but I don't know what my range for $\theta$ is supposed to be.
I don't need the answer, but if someone could explain what I need to do I would be appreciative. I'd be willing to even answer the question myself if I were told how I was supposed to do so.