Thin Plates with Constant Density (Calculus II)

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SUMMARY

The discussion focuses on finding the center of mass of a thin plate with constant density (δ) over the region bounded by the parabola y = x - x² and the line y = -x. Participants clarify the integral setup, correcting an algebra mistake in the integration limits and expressions. The correct integral to evaluate is from 0 to 2 of (2x - x²) dx, which accurately represents the area between the curves. The forum confirms that the formulas used are valid for calculating the center of mass in this context.

PREREQUISITES
  • Understanding of calculus concepts, specifically integration.
  • Familiarity with the concept of center of mass in physics.
  • Knowledge of parabolic equations and their graphical representations.
  • Ability to manipulate algebraic expressions and perform integration.
NEXT STEPS
  • Study the derivation of the center of mass formulas for two-dimensional shapes.
  • Learn about the application of double integrals in finding areas and volumes.
  • Explore the properties of parabolas and their intersections with linear equations.
  • Practice solving similar problems involving integration of bounded regions in calculus.
USEFUL FOR

Students in Calculus II, educators teaching physics or mathematics, and anyone interested in applying calculus to real-world problems involving center of mass calculations.

dm41nes
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Thank you in advance for the help!

Homework Statement



Find the center of mass of a thin plate of constant density (delta) covering the given region.
The region bounded by the parabola y= x - x2 and the line y= -x



Homework Equations



See attachment question 15 p1

The Attempt at a Solution



See attachment question 15 p2
 

Attachments

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I like your last solution on attachment 2 except...
Integral from 0 to 2 (x-x^2-x) dx is not Int (2x - x^2) dx
I think you just have an algebra mistake there.
 
Thank you, well it was x-x^2-(-x). So, I added the double negative to the other x. Thats how I was able to get 2x.


Are these forumulas a certified way to find the center of mass?
 

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