Problem from black book.

(1) Find the maximum value of \displaystyle \int^{y}_{0}\sqrt{x^4+(y-y^2)^2}dx for all 0\leq y\leq 1.

(2) Calculate maximum value of \displaystyle \int^{1}_{0}\bigg(f(x)\bigg)^3dx . Given that -1\leq f(x)\leq 1. and \displaystyle \int^{1}_{0}f(x)dx=0.