Hi I made an attempt at this problem but Im not quite sure if I have the correct procedure.

find the exact coordinates of the centroid.

$\displaystyle y = 100 - x^2$

$\displaystyle y = 0$

So the graphs intersect at x=-10 and 10

the centroid is $\displaystyle (\bar{x}, \bar{y})$

and $\displaystyle \bar{x}= \frac{M}{m}$

So $\displaystyle M = 2\int_0^{10} \! xf(x) \, dx.=$

$\displaystyle 2\int_0^{10} \! x(100-x^2) \, dx.=$

$\displaystyle 2[50x^2-\frac{1}{4}x^4]\bigg|_0^{10}=$

$\displaystyle 5000$

$\displaystyle m = 2\int_0^{10} \! f(x) \, dx.=$

$\displaystyle 2\int_0^{10} \! (100-x^2)\, dx.=$

$\displaystyle 2[100x-\frac{1}{3}x^3]\bigg|_0^{10}=$

$\displaystyle \frac{4000}{3}$

Therefore, $\displaystyle \frac{M}{m} = \frac{15}{4}$

and by symmetry the centroid is (15/4, 15/4) right??