I am looking through some questions and answers from earlier exams, and the following problem has the solution (a) and (e). The thing that bothers me is that the integration goes from $-1$ to $1$, thus from calculus is going to be $0$?
Problem: Let $p(x)$ be the interpolation polynomial to $f(x)=x^3$ on the nodes $x_0=-1, x_1=0, x_2=1$. Which statement is wrong?
Answers:
$\begin{align} &(a) \int_0^1f(x)dx=\int_0^1p(x)dx \\ &(b) \int_0^1p(x)dx=p(1/2) \\ &(c) \int_0^1p(x)dx=1/2(p(0)+p(1)) \\ &(d) \int_0^1p(x)dx=1/6(p(0+4p(1/2)+p(1)) \end{align}$
I know that (b) is a Midpoint rule, (c) is a Trapozoid and (d) is the Simpsons rule. My question is why does the intervals not go from $[-1,1]$? but from $[0,1]$? And therefore not the integration is not $0$?