Uncorrelated but not independent random variables
Solution 1:
Let $X$ be a standard normal random variable and let $Y = X^2$.
Then, since $E(X) = E(X^3) = 0$, we have $E(XY) = E(X^3) = 0 =E(X)E(Y).$
However, they are not independent:
$$P(0<X<1,Y>1) = 0 \neq P(0<X<1)P(Y>1)$$
For a simple discrete example, let $X_1$ and $X_2$ be independent random variables each taking values in $\{0,1\}$ with $P(X_i = 0) = P(X_i=1) = 1/2$. Let $X = X_1 + X_2$ and $Y = X_1 - X_2$.
We have $E(X) = 1$, $E(Y) = 0$ and $E(XY) = 0$. Hence, $E(XY) = E(X)E(Y)$.
But $P(X=0,Y=0) = 1/4 \neq P(X=0)P(Y=0) = 1/8.$