find the centroid of the region

[sareen]

Find the centroid of the region bounded by the curve y=1-x^2 and the x-axis.

sol) I believe that x=zero

what about y? which formula is used?? how to find the area?

 

[galactus]

Yes, the x coordinate of the centroid is 0 because of symmetry.

Find the area under the curve. $\displaystyle \int_{-1}^{1}(1-x^{2})dx=\frac{4}{3}$

$\displaystyle \overline{y}=\frac{3}{4}\int_{-1}^{1}\int_{0}^{1-x^{2}}ydydx$

You know $\displaystyle \overline{x}=0$. The above gives the y coordinate of the centroid.

If you could balance it on a pencil, this would be the point of balance.