Putnam 1989 B6

Let $(x_1, x_2,\,\ldots x_n)$ be a point chosen at random from the $n$-dimensional region defined by $0<x_1<x_2<\cdots < x_n<1.$ Let $f$ be a continuous function on $[0,1]$ with $f(1)=0$. Set $x_0=0$ and $x_{n+1}=1$. Show that the expected value of the Riemann sum $$\sum_{i=0}^n (x_{i+1}-x_i) f(x_{i+1})$$ is $\int_0^1 f(t)P(t) dt$, where $P$ is a polynomial of degree $n$, independent of $f$, with $0\le P(t)\le 1$ for $0\le t \le 1$.