I need to calculate the double integral of $f(x,y)=12\tan^{-1}\left(\frac{y}{x}\right)$ over a domain bounded by several lines.
I've graphed the lines, and they create a region that is a rectangle within a wedge between two circles in region 1. Ive also found the upper and lower limits of $r_2 = 3$ $r_1 = 1$ and $\theta_2= \frac{\pi}{3}$ and $\theta_1 = \frac{\pi}{6}$. In now stuck on how to get $12\tan^{-1}\left(\frac{y}{x}\right)$ into the $\cos$ and $\sin$ plugins. I know that you can have $12\tan^{-1}\frac{r\sin\theta}{r\cos\theta}$ is then $12arctan(\tan \theta) r \frac{dr}{d\theta}$..but im stuck on the next step.
Cheers
You're trying to evaluate $$\int_{\theta=\pi/3}^{\pi/6}\int_{r=1}^3 12\theta r\,dr\,d\theta$$ and as you said you first get $$\int_{\theta=\pi/3}^{\pi/6}12\theta\left.\frac{r^2}{2}\right\vert_{r=1}^3\,d\theta=\int_{\theta=\pi/3}^{\pi/6}6\theta(9-1)\,d\theta.$$ Then simplify and integrate just like you did with respect to $r$.