The correct mathematical expression for the local error for the trapezoidal rule is $$ \begin{equation*} \int_a^bf(x)dx -\frac{b-a}{2}\left[f(a)+f(b)\right]=-\frac{h^3}{12}f^{(2)}(\xi), \end{equation*} $$ and the global error reads $$ \begin{equation*} \int_a^bf(x)dx -T_h(f)=-\frac{b-a}{12}h^2f^{(2)}(\xi), \end{equation*} $$ where \( T_h \) is the trapezoidal result and \( \xi \in [a,b] \).