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Math Help - Centroid question

  1. #1
    Junior Member
    Joined
    Nov 2009
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    39

    Centroid question

    Find the x-coordinate of the centroid of the region bounded by the following curves: y=x^2,y=x+6.
    Hint: Make a sketch and use symmetry where possible.
    x-coordinate of Centroid:

    I got the wrong answer but I came up with 558/740.
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  2. #2
    Senior Member
    Joined
    Dec 2008
    Posts
    319
    I came up with a different answer, but we may start by finding the value of x for which y=x^2 and y=x+6 intersect:

    \begin{aligned}<br />
x^2&=x+6\\<br />
x^2-x-6&=0\\<br />
(x-3)(x+2)&=0.<br />
\end{aligned}

    Our limits of integration are therefore x=-2,3. giving us the area of the region:

    A=\int_{-2}^3 (x+6)-x^2\,dx.

    To find the x-coordinate of the centroid, we just calculate

    \bar{x}=\frac{1}{A}\int_{-2}^3 x((x+6)-x^2)\,dx.
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