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.
