Thread: Finding the coordinates of a centroid

1. Finding the coordinates of a centroid

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.

$y = 100 - x^2$
$y = 0$

So the graphs intersect at x=-10 and 10

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

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

So $M = 2\int_0^{10} \! xf(x) \, dx.=$
$2\int_0^{10} \! x(100-x^2) \, dx.=$
$2[50x^2-\frac{1}{4}x^4]\bigg|_0^{10}=$
$5000$

$m = 2\int_0^{10} \! f(x) \, dx.=$
$2\int_0^{10} \! (100-x^2)\, dx.=$
$2[100x-\frac{1}{3}x^3]\bigg|_0^{10}=$
$\frac{4000}{3}$

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

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

2. tried using this formula and I arrive at the same answer

$\bar{x} = \frac{1}{A}2\int_0^{10} \! x(100-x^2) \, dx.$

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

I know its obvious why, but am I doing this problem correctly?