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 $f(x_i)$.


Notation:
  • $a$ is the starting point;
    $b$ is the end point.
  • $n$ is the number of pieces in which the interval $[a,b]$ is subdivided.
  • $\Delta x = \displaystyle{\frac{b-a}{n}}$ is the size of each of those sub-intervals.  DO: Why?
  • $[x_{i-1}, x_i]$ is the $i$th interval; in particular $x_0=a, x_1=a+\Delta x, \ldots$, $x_i=a+i\Delta x,\ldots, x_n=b$.  DO: Why?
  • $x_i^*$ is any representative from the $i$th interval (usually the right endpoint, but could be the left, or midpoint, or any other value in the interval)
  • $f(x_i^*)$ is the height of the rectangle $R_i$ over the $i$th interval.
  • $f(x_i^*) \Delta x$ is the area of $R_i$.


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.

         

While each choice will give us different approximations, they will all give us the same answer at the limit.