Find the y-coordinate of the centroid of the semi-annular plane region by $\displaystyle 1 <= x^2 + y^2 <= 4$ and $\displaystyle y > 0$.

Printable View

- Jun 6th 2008, 11:03 AMkenshinofkinY-coordinate of a centroid
Find the y-coordinate of the centroid of the semi-annular plane region by $\displaystyle 1 <= x^2 + y^2 <= 4$ and $\displaystyle y > 0$.

- Jun 6th 2008, 11:28 AMTheEmptySet
The Mass of the plate is given by the integral

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

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

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

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

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

So your integal is

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

just distribute the x and integrate with a u sub

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

I hope this helps.

Good luck.