I'm reviewing this document:

http://ocw.mit.edu/courses/mathemati...18_03S10_i.pdf

The short story is that $\displaystyle w(t)$ is the weight function (solution to the associated homogeneous equation) of some generic equation $\displaystyle my'' + cy' + ky = f(t)$. We want to break the interval between $\displaystyle 0 < t < x$ into smaller intervals, $\displaystyle [t_{i}, t_{i+1}], i = 0, 1, 2, 3, ..., n$ and define $\displaystyle f_{i}(t) \begin{cases}f(t_{i}) \text{ if } t_{i} < t < t_{i+1}\\0 \text{ elsewhere}\end{cases}$ so that over the interval it's a constant function. This means that $\displaystyle f(t) \approx f_{0}(t) + f_{1}(t) + ... f_{n-1}(t)$.

I get everything until page 3 equation (8). If $\displaystyle w(t)$ is the weight function for $\displaystyle f(t)$, then I would think that for $\displaystyle f_{i}(t)$, the weight function would just be $\displaystyle w(t-t_{i})$ (restricted to the interval). I don't understand why we need to amplify this by $\displaystyle f(t_{i})\Delta t$.

(I forgot to include a part about Leibniz's Formula but I'm too tired.)