Measuring the area/volume/etc. of such a bounded region is a concept that lies at the heart of calculus and analysis. The keyword that captures the idea here is "definite integral". If you search for that, you'll find a plethora of resources that discuss its meaning and usage. Don't let the fact that it's a calculus topic fool you into thinking it's a very difficult thing to understand. It's actually quite intuitive, but it can be formulated to describe very, /very/ abstract things.
I think the best place to start the discussion would be to look for historical examples, such as Archimedes' proof (from around the 200s BCE) that any circle of radius 1 has area
. (Some URLs aren't allowed in posts, but work in comments, so I'll be sure to add them there.)
A few centuries later, mathematicians like Bernhard Riemann (mid-1800s)Â formulated a means of computing bounded areas by approximating it with the areas of rectangles. This is standard fare within calculus curricula across the world as far as I'm aware. As the concept is very intuitive, you can essentially consult the animations from the wiki page on Riemann integration. (I'll include that link too, if it's allowed.)
As calculus was further developed, it was discovered that areas such as the one in your problem, provided the function representing the area are "nice enough", can be computed by examining a different function called the antiderivative. Since you're working at an algebra II level, you won't have to worry about this for some time. The takeaway is that "nice enough" functions have properties that allow us to make the approximating process easy.
For especially "nice" functions, approximations may not even be necessary.
In the case of your problem, we can roughly approximate the area with a single rectangle (first attachment). You'll recall that areas of rectangles are easy to find.
We can get a better approximation by splitting up the region and using more rectangles (second attachment).
This is the idea behind Riemann integration. We can find a systematic way of constructing a sequence of rectangles whose areas can be summed, and this sum will converge to the area of the bounded region as the "partitioning" of the region (the actual splitting of the region into rectangles) is "refined" (more and more rectangles are used whose dimensions becomes closer and closer approximations of the bounded region). The animation included in the wiki page for Riemann integration illustrates exactly this process.
When I say "converge", that means we essentially consider the sum of an infinite number of rectangles. (The fact that an infinite number of numbers can add up to a finite number is itself an important notion that's discussed in breadth in any calculus course. You may also be interested in looking into the topic investigating what are called "limits".)
In short,
(note that
was replaced with
)
This sum is expressed in what's called sigma notation, which captures both the choice of the rectangles' dimensions:
and the infinite sum is often rewritten in a new notation specifically designed to denote definite integrals,
where
could be read as the "integral over the region
".
would capture the "height" of the region, while
is basically a means of capturing "width".
In this case, the area of the region is given by
and the value of this integral turns out to be
.
