Consider the area of the region bound by $y=f(x)$, the x axis, and joining vertical segments $x=a$ and $x=b$. Subdivide the interval $ a \leq x \leq b$ into n subintervals by means of the points $x_1, x_2,....x_{n-1}$, chosen arbitrarily. In each of the new intervals, choose points $\zeta _1, \zeta _2 ... \zeta_n$ arbitrarily. With $x_0 = a$, $x_n =b$ and $(x_k - x_{k-1}) = \Delta x_k$ this can be written as $\displaystyle\sum\limits_{k = 1}f (\zeta_k)\Delta x_k$, which represents the total area of all rectangles.
My question is, why?Especially the need for the arbitrary points $\zeta$ are unclear to me