In a text introducing basic composite quadrature rules (Simpson, Trapezoidal, ...) i read the following conceptual derivation for the general form of quadrature rules defined on $[0,1]$
$$\int_{a}^{b} f(x)\ dx \ \ \ \textbf{(1)}$$ $$= \sum_{i=0}^{N-1} \int_{t_{i}}^{t_{i+1}} f(x)\ dx $$ $$= \sum_{i=0}^{N-1} h_{i} \int_{0}^{1} f(t_{i} + h_{i}\sigma)\ d\sigma \ \ \ \textbf{(2)}$$ $$= \sum_{i=0}^{N-1} h_{i} \ Q(f(t_{i} + h_{i}))$$
where $[t_{i}, t_{i+1}]$ are subintervals of $[a,b]$ and $h_{i} = t_{i+1} - t_{i}$.
I understand the substitution of $x$, but wouldn't the sum over the integral in $\textbf{(2)}$ without multiplication of $h_{i}$ equal $\textbf{(1)}$?
Where does the multiplication of $h_i$ in $\textbf{(2)}$ come from?