Need help in understanding the formal definition of the limit.

Question

Consider the following definition given by the author of my book:

The definite integral of the function $f(x)$ on the interval $[a,b]$ is the limit of the integral sums $$\int_a^b f(x) dx = \lim \sum_{i=0}^{n-1} f(\xi_i) \Delta x_i \text{ when }\max |\Delta x_i|\to0.$$

Where $\xi_i\in[x_{i-1},x_i]$ and $\triangle x_i=x_i-x_{i-1}$. In order for the Reimann sum to be equal to the actual area under the curve the interval $\triangle x_i$ must become infinitesimaly small. But I don't understand how the notation used by the author conveys this idea. Please explain.

Answer 1

For each $i=1,...,n-1$, the notation $f(\xi_i)\Delta x_i$ can be interpreted as the area of a rectangle of base $\Delta x_i$ and height $f(\xi_i)$.

Thus, the notation $\displaystyle \sum_{i=0}^{n-1}f(\xi_i)\Delta x_i$ (sum of the areas of the $n-1$ rectangles) can be interpreted as an approximation to the actual area.

As you said, in order this sum to be equal to the actual area, the interval $\Delta x_i$ must become infinitesimally small (for all $i=1,...,n-1)$. This part is covered by the condition $\max|\Delta x_i|\to 0$.

Explicitly, the definition $$\int_a^bf(x)\;dx=\lim\sum_{i=0}^{n-1}f(\xi_i)\Delta x_i\text{ when }\max\Delta x_i\to 0$$ means

For all $\varepsilon >0$ there exists $\delta>0$ such that $$|x_i-x_{i-1}|<\delta\text{ for all }i=1,...,n-1\quad\Longrightarrow\quad \left|\sum_{i=0}^{n-1}f(\xi_i)\Delta x_i-\int_a^bf(x)\;dx\right|<\varepsilon$$

According to the previous interpretation, this means that if the base of all rectangles are small enough, then the difference between the actual area and the approximation is very small.