E(Y) = E(E(Y) | θ))
= E(P(Y) | θ)) since Y only takes the values 0 or 1
= E(θ) by the Bernoulli distribution
= ∫_{0}^{1}θp(θ)dθ
= ∫_{0}^{1}2θ^{2}dθ by part (a)
= 2/3
Note: this integral is analogous to summation in the discrete case, with probabilities equal to the areas of thin rectangles under the density function's graph: P(θ) = p(θ)dθ.