Solution
= Solution
From the marginal density,
$$
\mathbb E X=12\int_0^1x^2(1-x)^2\,dx=\frac25,
\qquad\mathbb E Y=\frac25.
$$
The mixed moment is
$$
\mathbb E[XY]=\int_0^1\int_0^{1-x}24x^2y^2\,dy\,dx
=8\int_0^1x^2(1-x)^3\,dx=\frac2{15}.
$$
Thus the <covariance> is
$$
\boxed{\operatorname{Cov}(X,Y)=\frac2{15}-\frac4{25}=-\frac2{75}.}
$$
The negative sign is consistent with the support constraint: a large value of one variable leaves a smaller available range for the other.