OK, I'm pretty sure this method is valid, avoiding double integrals.

First we need mass M.

$\displaystyle M = \int_a^b \rho(x) (g(x)-f(x)) dx$

Where in this case $\displaystyle g(x) = x$, $\displaystyle f(x) = x^2$, $\displaystyle \rho(x)=12x$, $\displaystyle a = 0$ and $\displaystyle b = 1$.

Then we write

$\displaystyle \bar x = \frac{1}{M} \int_0^1 x\rho(x)(g(x)-f(x)) dx$

$\displaystyle \bar y = \frac{1}{M} \int_0^1 \rho(x)\left(\frac{1}{2}\right)(g^2(x)-f^2(x)) dx$

I got these equations by combining

this reference (which seems to have a typo) and

this reference, with modifications.