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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) $.
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 8
Improper Integrals
Integration Techniques
Oregon State University
University of Michigan - Ann Arbor
University of Nottingham
Lectures
01:53
In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.
27:53
In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.
13:54
Astronomers use a techniqu…
16:14
01:12
Suppose that the density a…
03:29
To see in principle how as…
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
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