Integration Engineering Problem

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SUMMARY

The discussion centers on calculating the mass of a cylinder with variable density, defined as K times the distance from the base. The cylinder has a length of 5 units and a diameter of 2 units. The mass is derived using integration, leading to the formula M = (25Kπ)/2. The approach involves slicing the cylinder into infinitesimal pieces and summing their masses through a definite integral.

PREREQUISITES
  • Understanding of integral calculus, specifically definite integrals.
  • Familiarity with the concept of variable density in physics.
  • Knowledge of geometric properties of cylinders, including volume and cross-sectional area.
  • Basic proficiency in setting up and solving integrals related to physical problems.
NEXT STEPS
  • Study the derivation of mass using integrals in variable density problems.
  • Learn about the applications of definite integrals in calculating physical properties of solids.
  • Explore the concept of slicing in calculus and its relevance to volume and mass calculations.
  • Investigate the relationship between density, mass, and volume in varying density scenarios.
USEFUL FOR

Students in physics or engineering, particularly those studying mechanics and material properties, as well as educators looking for practical examples of integration in real-world applications.

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Homework Statement



A cylinder of length 5 and diameter 2 units is constructed such that the density of the material comprising it varies as the distance from the base. If K is a proportionality constant such that the density is given by K * distance from base, use an integral to show that the
mass of the cylinder is given by : M =25Kpi/2

Homework Equations



Area of Circle : A = pi* r^2

The Attempt at a Solution



Fairly stuck with this question. I am not even sure where to begin with this. I am certain it isn't a very difficult problem but I am just not sure where to begin with it. Its going to consist of a definite integral somewhere along the line.
I understand that questions don't get answered without some sort of an attempt but I am stumped at where to begin with this.
 
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Try slicing up the cylinder in little pieces of width dz.
If you take such a slice, at height z to dz, you can assume that its density is constant for sufficiently small values of dz. What is the mass then?

When you add this up for all slices, that will give you the definite integral you are looking for.
 

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