I want to calculate $$\operatorname{E} \left[ \int_0^1{W(t)dt \cdot \int_0^1{t^2W(t)dt}} \right].$$
I discovered that the first integral is $\operatorname{N}(0, \frac{1}{3})$ but I don't know how to get the other one and the full answer of their multiplied expectation.