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A planet is in the shape of a sphere of radius $R$ and total mass $M$ with spherically symmetric density distribution that increases linearly as one approaches its center. What is the density at the center of this planet if the density at its edge (surface) is taken to be zero?

Calculus 3

Chapter 15

Multiple Integrals

Section 7

Triple Integrals in Cylindrical and Spherical Coordinates

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Lectures

04:18

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. The area above the x-axis adds to the total.

26:18

In mathematics, a double integral is an integral where the integrand is a function of two variables, and the integral is taken over some region in the Euclidean plane.

08:00

Density of center of a pla…

01:01

Find the mass of a spheric…

02:01

The density inside a solid…

in this problem. We are analyzing a sphere, and we have to arbitrarily understand its radius and its density and its mass. And we're told we have to use all of this knowledge as well as our knowledge of integration, to find the mass of a thin strip of our sphere in order to find this density we're looking for. So at first let's start this problem by denoting density, as did not well, D of our is going to be equal to D not times Big ar minus little are all over big are where big R is the radius of the earth and little are is a radial distance. So what we need to do to find the mass of this thin strip and later density is to understand what the volume of that thin strip of this fear is going to be. Well, it's going to be four or three pi times R plus d r cubed minus r cubed and we can simplify that to get four pi r squared D R. Well, at this point, the density is going to be equal to D not times a big r minus. The little are all over our big are to be exact. Well, we want to know the mass of this strip because what is density? We need to know volume and mass and we have volume and we're going to have density so we can now determine how to find the mass of the strip. The mass would be equal to D not times Big ar minus little are times four pi r squared d r over our well we can now use integration to solve for d not will take the integral from zero to big art of D not times big ar minus little are times four pi r raised to over big r N d r And when we do those steps to solve this integral will get pi times do not times big r cubed over three and that's going to be equal to the mass of this strip in our sphere. Well, we can rearrange this to get that our density did not equals three m m being our mass uber pie big r cubed. And that's the answer to this problem. That's how we would find the density of this fear that we're investigating. So I hope this helped you understand a little bit more about how we can use our knowledge. Ah, volume mass and density to figure out the density of a sphere that we're investigating using integral techniques.

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