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Problem 66 Easy Difficulty

Astronomers use a technique called stellar stereography to determine the density of stars in a star cluster from the observed (two-dimensional) density that can be analyzed from a photograph. Suppose that in a spherical cluster of radius $ R $ the density of stars depends only on the distance $ r $ from the center of the cluster. If the perceived star density is given by $ y(s) $, where s is the observed planar distance from the center of the cluster, and $ x(r) $ is the actual density, it can be shown that
$$ y(s) = \int_s^R \frac{2r}{\sqrt{r^2 - s^2}} x(r)\ dr $$
If the actual density of stars in a cluster is $ x(r) = \frac{1}{2} (R - r)^2 $, find the perceived density $ y(s) $.

Answer

$$y(s)=R^{2} \sqrt{R^{2}-s^{2}}-\frac{2}{3}\left(R^{2}-s^{2}\right)^{3 / 2}+R s^{2} \ln \left|\frac{s}{R+\sqrt{R^{2}-s^{2}}}\right|$$

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Video Transcript

So for the following problem is going to be discussing improper intervals. We know that astronomers used a technique called stellar stereo graffiti to determine the density of stars and a star cluster from the observed two dimensional density analyzed from a photograph. If we have a perceived star density given by Y. S. Whereas as the observed planer distance from the center of the cluster, an ex of our is the actual density. We see how wives can be written. It's going to be an integral and this is going to be a case of an improper integral. And we're told that the actual density of the stars in the cluster is X of R equals one half R minus R squared. So in finding the perceived density, what we're going to do is plug in that one half r minus R squared and we can simplify the terms further. Once we do that, we'll end up getting our final answer to being R squared times the square root of R squared minus x squared minus two thirds times R squared minus escort to three halves plus R. S squared times the natural log of the absolute value of S over our plus the square root of R squared minus X squared. So definitely a lot of work to do. But ultimately know that we're going to be finding wives. Once we know our value of X of our and we can use improper integral techniques to find this