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Finding the centroid of the region

  1. Feb 1, 2013 #1
    Hi

    VVVVVV

    Find the centroid of the region bounded by the curves y = 2x - 4 , y = 2 Sqr x, and x = 1. Make a sketch of the region.
     
  2. jcsd
  3. Feb 1, 2013 #2

    rollingstein

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    Make an attempt?
     
  4. Feb 1, 2013 #3
    I did not try ..because I am not familiar with the way of solving
     
  5. Feb 1, 2013 #4

    haruspex

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    Did you at least manage to sketch the region? You have three lines. Find out where they meet.
     
  6. Feb 1, 2013 #5
    yeah I did sketched the three curves then ?..
     
  7. Feb 1, 2013 #6

    haruspex

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    So where do they intersect? What formula do you know for finding a coordinate of a centroid?
     
  8. Feb 1, 2013 #7
    they intersect between 1 and ( a unknown point intersection between 2 Sqrt x and x = 1 )
     
  9. Feb 1, 2013 #8
    also, there is a triangle region under x-axis between 1 and 2 but I think it is not included in the intersection
     
  10. Feb 1, 2013 #9
    how can I know the second point of the limit integration ?!
     
  11. Feb 1, 2013 #10
    I got the answer :D >>

    thank u : )
     
  12. Feb 1, 2013 #11

    HallsofIvy

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    I am puzzled by this. Why in the world would you be given a homework problem like this if you had never been given instruction in these and your text book has nothing on it? You are taking Calculus are you not? And every text I have seen has the formulas for "centroid". You also seem to be saying that you cannot solve a simple quadratic equation.

    In any case, The line x= 1 intersects [itex]y= 2\sqrt{x}[/itex] at (1, 2) and the line y= 2x- 4 at (1, -2) and forms the left boundary. The line y= 2x- 4 and [itex]y= 2\sqrt{x}[/itex] intersect when [itex]y= 2x- 4= 2\sqrt{x}[/itex]. Divide by 2 to get [itex]x- 2= \sqrt{x}[/itex] and square both sides: [itex](x- 2)^2= x^2- 4x+ 4= x[/itex] or [itex]x^2- 5x+ 4= 0[/itex]. That factors easily: (x- 4)(x- 1)= 0. Either x= 1 or x= 4. The point (1, -4) is an "extraneous" root- it is not on [itex]y= 2\sqrt{x}[/itex]. So the last vertex, the intersection between [itex]y= 2\sqrt{x}[/itex] and y= 2x- 4, is at (4, 4).

    The area of that region is given by the single integral
    [tex]\int_1^4 2\sqrt{x}- (2x- 4)dx[/tex]
    which could also be done by the double integral
    [tex]\int_1^4\int_{2x- 4}^{2\sqrt{x}} dydx[/tex].

    I give that double integral (which easily integrates with respect to y to give the first integral) because it is needed for the centroid.

    The x coordinate of the centroid is given by the integral of x over that region
    [tex]\int_1^4\int_{2x-4}^{2\sqrt{x}} x dydx= \int_1^4 x(2\sqrt{x}- (2x-4))dx[/tex]
    divided by the area and

    the y coordinate of the centroid is given by the integral of y over that region
    [tex]\int_1^4\int_{2x-4}^{2\sqrt{x}} y dy dx[/tex]
    divided by the area.
     
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