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Kelly throws a dart at a circular dartboard of radius $3$ feet. Let X and Y denote the location where the dart lands. Assume that $−3 ≤ X ≤ 3$ and $−3 ≤ Y ≤ 3$ and $X^2 + Y^2 ≤ 9$, i.e., the dart lands on the dartboard. Moreover, assume that the dart’s location is Uniform on the dartboard, i.e., $f (x, y) = 1/(9π)$ if $x, y$ are on the dartboard, i.e., $x^2 + y^2 ≤ 9$

Let $D = \sqrt{X^2 + Y^2}$ be the distance from the dart to the center of the dartboard. Find $E(D)$.

I need help finding the bounds of integration to find $E(D)$ can someone lend me a hand?

David G. Stork
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    What do you think they are? – Joe Apr 04 '21 at 23:58
  • I was thinking it would be $r = 0$ to $3$ and $\theta = 0$ to $2\pi$ with an double polar integral, but the issue is that I'm given $\sqrt{X^2+Y^2}$ and that's not polar... –  Apr 05 '21 at 00:05
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    If you want to use polar coordinates, you could express $D$ in polar coordinates too. Don’t forget the Jacobian for change of variables. – Joe Apr 05 '21 at 00:08
  • so please tell me if I got this right $E(D)$ would be would be $\int_0^3 \int_0^{2\pi} r^2\cfrac{1}{9\pi}drd\theta$? –  Apr 05 '21 at 00:12
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    Yes............ – Joe Apr 05 '21 at 00:14
  • I just realized I switched up the integrals by accident in the above. But I get 2 as my final answer which double checking with a calculator seems right. You said earlier "If you want to use polar coordinates", is there an alternative and easier way to do this question? –  Apr 05 '21 at 00:19
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    I think polar is easier, but otherwise I would say $X \in [-3,3]$ and $Y \in [-\sqrt{9-X^2},\sqrt{9-X^2}]$. Also, this is a related post - https://math.stackexchange.com/questions/2438705/expected-distance-btw-random-point-inside-the-circle-and-its-centre – Joe Apr 05 '21 at 00:22
  • Sorry ignore that last comment. I'm just trying to find out what I unlocked on stack exchange –  Apr 05 '21 at 00:46
  • Welcome to SE. If you write up your answer and explain it well, I’ll upvote it. – Joe Apr 05 '21 at 01:36

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