Thin Plates with Constant Density (Calculus II)

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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
  • Handwritten formulas for the centroid of a plane region: x-bar = (1/A) times the integral from a to b of x[f(x) - g(x)] dx, y-bar = (1/A) times the integral of (1/2)[f(x)^2 - g(x)^2] dx, and A = integral from a to b of [f(x) - g(x)] dx, with the note 'to solve this? The region bounded by y = x - x^2 and y = -x'.
    question 15 p1.png
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  • Handwritten calculus work finding the centre of mass of a plate bounded by y = x - x^2 and y = -x, with a sketch of the region, intersection points x = 2 and 0, moment integrals, area 4/3, and final answer (2, -6/7).
    question 15 p2.png
    36.6 KB · Views: 865
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?