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Thread: Centroid question

  1. #1
    Junior Member
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    Nov 2009
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    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
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    I came up with a different answer, but we may start by finding the value of $\displaystyle x$ for which $\displaystyle y=x^2$ and $\displaystyle y=x+6$ intersect:

    $\displaystyle \begin{aligned}
    x^2&=x+6\\
    x^2-x-6&=0\\
    (x-3)(x+2)&=0.
    \end{aligned}$

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

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

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

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