When talking about any mathematical quantity, it's important to keep
track of three different questions:

What is it?

How do you compute it?

What is it good for?

This is especially true for integration, where the three questions have very
different answers:

The integral $\int_a^b f(x) dx$ is the limit of a sum:
$\displaystyle{\lim_{n\to\infty} \sum_{i=1}^n f(x_i^*)\Delta x}$.

We compute integrals with the Fundamental Theorem of Calculus:
$\int_a^b f(x) dx = F(b)-F(a)$, where $F'(x)=f(x)$.

We use integration to compute
the total amount of stuff in the interval
$[a,b]$, where stuff could be area under a curve,
distance traveled, volume of a solid of revolution, arc length
of a curve, or a host of other applications.

Of these applications, the easiest to understand is area under a curve,
which is why we used area to introduce integrals. Likewise, we will use
volume to introduce double integrals. But remember: ordinary integrals are
about much more than area, and double integrals are about much more
than volume.

In the following video, we review the ideas and definitions
of one-dimensional integrals. Not how to compute an integral, but
what it is and what it's good for. On the next page, we'll see how
the exact same ideas work in two and three dimensions. We'll tackle
'how do you compute it?' in the next learning module.

Remember the main idea of integration:
The whole is the sum of the parts.
Integration is a procedure for computing bulk quantities like area,
volume, mass, distance, moment of inertia, and wealth.
If $f(x)$ is the density of a quantity on interest, then

Break the interval $[a,b]$ into $N$ pieces, each of size
$\Delta x = (b-a)/N$. Label the pieces by an index $i$ that goes
from 1 to $N$.

For each $i$, pick an arbitrary
sample point $x_i^*$ somewhere in the $i$-th interval.

Approximate the amount of stuff in the $i$-th piece as
$f(x_i^*) \Delta x$.

Approximate the total amount of stuff as
$\displaystyle{\sum_{i=1}^N f(x_i^*) \Delta x}$.

To get an exact answer, take a limit as $N \to \infty$:

These ideas are illustated with the standard `area under a curve' example.
The video on the right shows how this area can be approximated by adding up
narrower and narrower rectangles, with $x_i^*$ being the left endpoint.
(You may need to hit the 'replay' button
a few times, since it's only 6 seconds long.)