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Math Help - Y-coordinate of a centroid

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    Y-coordinate of a centroid

    Find the y-coordinate of the centroid of the semi-annular plane region by 1 <= x^2 + y^2 <= 4 and y > 0.
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    Quote Originally Posted by kenshinofkin View Post
    Find the y-coordinate of the centroid of the semi-annular plane region by 1 <= x^2 + y^2 <= 4 and y > 0.
    The Mass of the plate is given by the integral

    \int_{-2}^{2}(\sqrt{4-x^2}-\sqrt{1-x^2})dx

    but you do need calculus to find the mass you can use geometery

    m=\frac{1}{2}\pi (2)^2-\frac{1}{2} \pi (1)^2=\frac{3\pi}{2}

    M_y=\int_{a}^{b}x[f(x)-g(x)]dx

    Where f(x)=\sqrt{4-x^2} \\\ g(x)=\sqrt{1-x^2}

    So your integal is

    \int_{-2}^{2}x(\sqrt{4-x^2}-\sqrt{1-x^2})dx

    just distribute the x and integrate with a u sub

    the centroid is given by \frac{M_y}{m}

    I hope this helps.

    Good luck.
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