Riemann Sum Notation

The total area under $y=f(x)$ on an interval is approximated by $$\sum_{i=1}^n \,f(x_i^*)\, \Delta x,$$ which is the sum of the areas of $n$ rectangles.  This sort of expression is called a Riemann Sum.  We use the Greek letter sigma ($\Sigma$) to mean sum.  The expression
$\displaystyle{\sum_{i=1}^n (\hbox{formula involving $i$})}$ means "plug $i=1$ into the formula, then plug in $i=2$, all the way up to $i=n$, and add up the terms."  Thus:

$$\sum_{i=1}^n \,f(x_i)\, \Delta x=\,f(x_1)\, \Delta x +\,f(x_2)\, \Delta x+\,f(x_3)\, \Delta x+\cdots+\,f(x_{n-1})\, \Delta x+ \,f(x_n)\, \Delta x.$$

In the video, keep referring to the left side for a list of symbols. You will need to learn the meaning of, and how to find, the values represented by $a,b,n,\Delta x,x_i$ and $f(x_i)$.

The exact area is the limit of the Riemann sum as $n \to \infty$. Notice that we could use the left endpoint $x_{i-1}$, the right endpoint $x_i$, the midpoint $\frac{x_{i-1}+x_i}{2}$, or any other representative point.