Solution
= Solution
For the <composite trapezoidal rule>, the mesh width is $h=1/N$ and the nodes are $x_n=n/N$. The endpoint weights are halves and the interior weights are ones, so
$$
\boxed{T_N=\frac1N\left[\frac{3(0)^2}{2}+\sum_{n=1}^{N-1}3\left(\frac nN\right)^2+\frac{3(1)^2}{2}\right]
=\frac3{N^3}\sum_{n=1}^{N-1}n^2+\frac3{2N}.}
$$
This formula is an approximation to the <definite integral>, not an exact equality with it for finite $N$.