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Integration Application: Linear Density Question


Bwave

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I recently was marked down on my homework when solving a problem relating to linear density. I understand the process of computing the mass, but in my mind it doesn't make sense. I would appreciate if someone could explain it to me.

 

The problem similar to as follows:

Calculate the mass of a 3 meter rod whose linear density is calculated using the formula p(x) = sin(3x)+5 kilograms per meter.

 

I understand you setup an integral w.r.t. x but in my mind the problem is missing a component. Wouldn't the mass of the rod vary depending on the radius of the rod? I understand that the density is linear, but if the rod was 30 meters in radius, wouldn't it be greater than if the rod had a .1 meter radius?

 

My final answer ended up including a variable R because I thought the mass would change depending on the thickness. I simply found the cross-sectional area of the rod which was just a circle: piR^2. I then multiplied the circle times the density equation. I then computed the volume by adding up all the circles from the beginning of the rod to the end of the rod. It ended up being 2piR^2 times the integral from 0 to 3 of (x(sin(3x)+5))dx. The correct answer was simply a value and I am a little bit confused by this.

 

Could anyone provide a reasonable explanation as to why the radius is not taken into account in linear density problems?

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