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I have the following density function: $f_{(x,y)}(t,s) = \frac{1}{3\pi}$ for $1\le(t-2)^2+s^2\le4$ and else $f_{(x,y)}(t,s) =0$.

I need to find $f_y(y) = \int{f(x)dx}$, but I having trouble to find the range of the integral, even though i draw it.

I thought about changing the variables as this:

$x=rcos(\phi)+2, y=rsin(\phi)$.

Even though it maybe will make the integral more simple, i'm still find it difficult to find the range.

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