The Logical Framework For Definite Integrals
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In this lesson we are going to go through the logical framework that defines a definite integral. There is a lot of logic that you need to fully understand what is going on here. (Sorry about that.) However, in this lesson we have tried to strip out the logic and present the details through statements and a picture accompanying every statement. This way you can get 100% of the intuition without having to go through the logic. As always we provide a link to the problem that details the logic. At the end we will summarize everything with the two pictures we want you to keep in mind. Finally, by going through this you will find that the logical framework can be understood with some simple pictures just like we pictured integration as upper and lowers sums of rectangles. This, however, is the true logical framework and will allow us to prove a lot later in other lessons and problems.
The Logical Framework of Definite Integrals  

Assume we have a bounded function $f(x)$ on the closed interval $[a,b]$.  
We can show that the set of all lower sums has a least upper bound. We call it $\phi$.  
We can show that the set of all upper sums has a greatest lower bound. We call it $\Phi$.  
We can show that <Sign in to see all the formulas>. So now in general we know that for any two partitions $P$ and $Q$,<Sign in to see all the formulas> 

Next we can show that<Sign in to see all the formulas>and <Sign in to see all the formulas>This means that when we put more and more points into a partition the value of the upper sums get smaller and approach $\Phi$. Similarly, the lower sums will get bigger and bigger and approach $\phi$. 

Doesn't it seem intuitive that the closer $\phi$ and $\Phi$ are to each other the closer we can get $U(P)$ and $L(P)$ with more points in a partition? In fact, we can show that if <Sign in to see all the formulas>, then for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that for all partitions $P$, <Sign in to see all the formulas>. In our original lesson on upper and lower sums we argued intuitively that we want to view putting in more and more points to get the upper and lower sums closer to one another and thus arrive at the value of <Sign in to see all the formulas>. We can actually prove a stronger result that says we can get $U(P)$ and $L(P)$ as close to each other as we need if and only if <Sign in to see all the formulas>.  
Now when <Sign in to see all the formulas> something else almost magical happens. We can show there is a unique number $I$ such that for all partitions $P$, <Sign in to see all the formulas>. (We can also prove the stronger if and only if version here.) But this is exactly our original definition of a definite integral! In other words,<Sign in to see all the formulas>This can be written more succinctly as: The bounded function $f(x)$ on $[a,b]$ is integrable on $[a,b]$ if and only if <Sign in to see all the formulas>. This is oftentimes referred to as the Darboux Integrability Theorem. 
Now we are ready to reveal the most important fact of this lesson. The last two points above are very strong mathematical statements. With them we can actually show that our definition of a definite integral can have three equivalent forms:
Three Equivalent Definitions of the Definite Integral 

When we prove one of these is true then we know the other two are true! 
Ok, we promised we would give you the two pictures to keep in your head so you could have a good intuition of what is going on. We recommend memorizing and understanding these two pictures from all the pictures above:
Finally, finding the value of a definite integral as we have  by calculating upper and lower sums and using partitions with smaller and smaller norms to find <Sign in to see all the formulas>  is oftentimes referred to as Darboux Integration the upper and lower sums referred to as Darboux sums. We mention this because some books and websites might immediately call them Darboux sums and Darboux integrals. It's also important to note that there is another kind of integration called Riemann Integration. Don't worry too much about this because we show here that Darboux integration and Riemann integration are the same thing and give the same number for <Sign in to see all the formulas>. We just want you to be aware of other ways these concepts can be labelled.