We develop the algebraic techniques to actually compute the perfect approximation of area under the curve.

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We’ve seen that we can get a perfect estimation of the area under a curve by taking a limit of the Riemann Approximation. This is remarkably useful, but the notation can be a little bulky. For that reason we introduce a different and more condensed notation here.

The perfect approximation of the area under a curve is referred to as a Definite Integral, and has it’s own condensed notation (there’s a little more to a definite integral as well, but we’ll discuss that when it comes up). In particular, our established formula to compute the exact signed area between and the -axis on the interval , is currently notated as:

Phew, that is... a lot. There is a lot to keep track of there, many different letters and the order of a sum along with the limit (and the order can matter as to which one happens when, but that is outside of the scope of this class). Not to mention it just takes up so much room! Remember... mathematicians are a lazy group by nature, so although the above is the correct definition of the perfect approximation of the signed area, mathematicians created a vastly more condensed way of writing it to save time and energy... not mention ink (or chalk).

The definite integral notation for the same expression as the above is expressed as:

Much. Better. Again, this is really just a cosmetic change (at least initially) - this is just a way to convey that same “limiting process of rectangle approximations of the area under the curve” without writing nearly so much.

1 : The main reason to use the style notation for definite integrals, rather than the notation is...
Because they are different things, the notation is designed for finitely many rectangles, whereas the notation is for infinitely many. Because the notation is too confusing and hard to keep track of things like an index, so we avoid using it whenever we can. It is (currently) just a convenience choice - we use the notation because it takes less time/effort/ink to write everything down, and we know that it’s really the same as the notation. To give us more stuff to memorize.

There is another reason to use this specific notation (eventually) - which has to do with a surprising relationship between the area under a curve and antiderivatives which will make this notation much more convenient (not to mention much easier to compute).

So, how do we read this notational shorthand? And where did it come from?

The integral sign was first used by Leibnitz and was actually an elongated “S”, which he used as a shorthand for “sum”. The left boundary of the region goes below the symbol, and is referred to as the “lower bound” and the right boundary of the region goes above the symbol, and is referred to as the “upper bound”.

The Definite Integral is so-called definite (as oppose to indefinite as we will see in the future) precisely because it has boundaries for the region. In short, it has well defined/finite (definite) boundaries/edges for the area.