Question 8
Let $f(x)=x^2(1-x)^2.$ What is the value of the sum
$$
\begin{gathered} f\!\left(\tfrac{1}{2019}\right)-f\!\left(\tfrac{2}{2019}\right) \\ {}+f\!\left(\tfrac{3}{2019}\right)-f\!\left(\tfrac{4}{2019}\right) \\ {}+\cdots+f\!\left(\tfrac{2017}{2019}\right) \\ {}-f\!\left(\tfrac{2018}{2019}\right)? \end{gathered}
$$
$$
\begin{gathered} f\!\left(\tfrac{1}{2019}\right)-f\!\left(\tfrac{2}{2019}\right) \\ {}+f\!\left(\tfrac{3}{2019}\right)-f\!\left(\tfrac{4}{2019}\right) \\ {}+\cdots+f\!\left(\tfrac{2017}{2019}\right) \\ {}-f\!\left(\tfrac{2018}{2019}\right)? \end{gathered}
$$