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:

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. This is why we normally just use the right
endpoint $x_i$.