Use polar coordinates in $\Bbb{R^2}$ to evaluate $$\iint_{R} \frac{x^2}{x^2 +y^2} \,dx\,dy$$ where R is the region between the concentric circles of equations $x^2 +y^2=a$ and $x^2 +y^2=b$ with $a<b$ and $(x,y)$ are Cartesian coordinates in $\Bbb{R^2}$
So I know the region is the area between the smaller circle $$x^2 +y^2=a$$ and the bigger circle $$x^2 +y^2=b$$ but i'm not sure how to get limits and be able to evaluate it when I don't have any values for a and b which is throwing me off.
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In polar coordinates, the integration region is parametrized as
$$ \left\{x = r\cos\left(\theta\right), y = r\sin\left(\theta\right)\right\}, \quad 0\leq\theta\leq 2\pi, a\leq \rho \leq b. $$
Changing the variables gives
$$ dxdy = rdrd\theta. $$
$$ \iint_R\frac{x^2}{x^2+y^2}dxdy = \int_0^{2\pi}\int_a^b\frac{r^2\cos^2\left(\theta\right)}{r^2\cos^2\left(\theta\right) + r^2\sin^2\left(\theta\right)}rdrd\theta = \int_a^brdr\int_0^{2\pi}\cos^2\left(\theta\right)d\theta $$
The region you are integrating over can be written as $[a,b] \times [0, 2 \pi )$ when you are transforming to polar coordinates. The radius $r$ ranges between $a$ and $b$ while the angle $\phi$ goes around the whole circle, hence the interval from $0$ to $2\pi$.
Now you can transform $$(x,y)=(r \cos(\phi), r \sin(\phi))$$ and with the Jacobian determinant being $J=r$ for two-dimensional polar coordinates we get the integral $$\int_a^b \int_0^{2 \pi} r \cos^2(\phi) \ d \phi dr$$ which can be evaluated relatively easy when first integrating with respect to $\phi$ and then with respect to $r$. The antiderivative of $\cos^2(x)$ is $\frac{\cos\left(x\right)\sin\left(x\right)+x}{2}$. Plugging in $2\pi$ and $0$ gives us $\pi$ as the value for the first integral. Now we just evaluate $$\pi \int_a^b r \ dr$$ which is $\frac{\pi}{2}(b^2-a^2)$