# Y-coordinate of a centroid

• Jun 6th 2008, 12:03 PM
kenshinofkin
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 $y > 0$.
• Jun 6th 2008, 12:28 PM
TheEmptySet
Quote:

Originally Posted by kenshinofkin
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 $f(x)=\sqrt{4-x^2} \\\ g(x)=\sqrt{1-x^2}$

$\int_{-2}^{2}x(\sqrt{4-x^2}-\sqrt{1-x^2})dx$
the centroid is given by $\frac{M_y}{m}$