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.
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? 🙂
Best Answer
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 $r$.