<s>[INST] multiple integral finding limit integration question : 
$\iint_D \ln(x^2 + y^2)dxdy$ ,
 domain : $4\le x^2+y^2 \le 9$ 
im having hard times to find the limit for integration  because it is in polar coordinates, i got that $2 \le r\le 3$ but i dont know how to find the $\theta$ limit. 
and also i dont know the graph actually, is it the same as logarithm graph? is it better to draw the graph? is there another way beside drawing the graph? thanks!
 [/INST] You make the change of variables: 
$$x=r\cos{\theta},y=r\sin{\theta}.$$
Hence: 
$$4\le x^2+y^2 \le 9 \Rightarrow 2^2\le r^2\le 3^2 \Rightarrow 2\le r\le 3.$$
The domain is the region between the cocentric circles with the center at the origin and the radii $2$ and $3$, therefore:
$$0\le \theta \le 2\pi.$$
Note: If the domain was $4\le x^2+y^2\le 9, x\ge 0, y\ge 0$, then it would be the region between the concentric circles with the center at the origin and the radii $2$ and $3$ that lies in the first quadrant, implying $0\le \theta \le \frac{\pi}{2}$.

The formula of double integration in polar coordinates is:
  $$\iint_{D} f(x,y)dxdy = \int_{\theta_1}^{\theta_2} \int_{r_1}^{r_2}f(r\cos{\theta},r\sin{\theta})\cdot Jdrd{\theta},$$
  where $J$ is the Jacobian:
  $$J=\begin{vmatrix}
        x_r & x_{\theta} \\
        y_r & y_{\theta} \\
        \end{vmatrix} = \begin{vmatrix}
        \cos{\theta} & -r\sin{\theta} \\
        \sin{\theta} & r\cos{\theta} \\
        \end{vmatrix} = r.$$

Hence the integral:
$$\iint_{4\le x^2+y^2\le 9} \ln(x^2+y^2)dxdy =\int_0^{2\pi} \int_2^3 \ln{r^2}\cdot rdrd{\theta}=$$
$$\left(\int_0^{2\pi}d{\theta}\right) \left(\int_2^3 \underbrace{\ln{r}}_{u} \cdot \underbrace{2rdr}_{dv}\right)=(2\pi)\left(\ln r \cdot r^2 \bigg{|}_2^3-\int_2^3 r^2\cdot \frac1r dr\right)=$$
$$(2\pi)\left(9\ln 3-4\ln 2-\frac52 \right)\approx 28.996.$$
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