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I have an integral $\int_D\,\frac{1}{x^2+y^2}dxdy$ which I should integrate over $D$.

$D$ is limited by $1 \leq x^2 +y^2 \leq 4$ and $x \geq 0, y \leq 0$

I have plotted the limits and the integration should be between the circles in the down right quadrant. enter image description here

Update: after hint of polar coordinate, limits and that $r$ should be $\frac{1}{r}$:

$\int_D\,\frac{1}{x^2+y^2}dxdy = \int_{\frac{3\pi}{2}}^{2\pi}\,\int_1^2{}\,\frac{1}{r}drd\theta$

finally correct? :)

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    Thanks for the polar hint and very much thanks for the riminder that angels are going counter clockwise I had a smaler blackout it seems :)2012-10-01

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Hints:

$\dfrac{1}{x^2+y^2}=\dfrac{1}{r^2}$

$\dfrac{1}{x^2+y^2}r=\dfrac{1}{r}$

See this question.

ADDED. Your update is now correct: the integrand in polar coordinates becomes $1/r\;$ after the multiplication by $r$.

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    @Farmor You are welcome.2012-10-01