Why can you transition from a Riemann Sum to an Intergal?

Question

I understand that the integral is defined through a Riemann Sum, as in $$\int_a^bf(x)\,dx=\lim_{n\to\infty}\sum_{i=0}^{n-1}f\left(a+i\cdot\frac{b-a}{n}\right)\frac{b-a}{n}$$ But I don't understand how this definition is made. I understand the structure of the Riemann Sum which is summing up infinetely many infinietely thin rectangles to approximate the area under a curve, but I don't understand why this turns into an integral and then can be evaluated the way an integral is. Any help is appreciated.

Answer 1

The expression $\int_a^b f(x)\,dx$ itself has no meaning before it is defined. The limit on the right gives a meaning to the expression on the left.

"... then can be evaluated the way an integral is". I assume that you are talking about the fundamental theorem of calculus: $$ \int_a^b f(x)dx=F(b)-F(a) $$ where $F$ is an antiderivative of $f$. That is proved (ultimately) using the definition of the expression "$\int_a^b f(x)\,dx$".

Answer 2

Perhaps a good way to think about that formula is the following:

There exist some functions f defined over the interval [a,b] which have the property that the limit on the right hand side exists and the sequence of sums converges to that limit. In the case where that happens we will name that limit, or call that limit by the symbology on the left hand side.

So then in the Analysis class or Advanced Calculus class, (depending on how the classes are named, in the old days and still in some places it goes by the name Calculus), you will prove that these limits exist for various families of functions, and you will also prove what is known as the fundamental theorem of calculus.

When you say evaluation you are applying the fundamental theorem of calculus and some of the other theorems which you will prove in the Analysis or the Advanced Calculus class.

So when you say:

"but I don't understand why this turns into an integral and then can be evaluated the way an integral is."

I interpret that to mean that you are not confused as to what the evaluation rules are and how to evaluate per se, but rather why you are allowed to use those rules to evaluate.

The answer to that query is the subject matter of what could be a first Analysis course and exceeds the length and detail allowed in this format. I encourage you to look into it further as time allows.

If time is an issue and you are trying to get through your current course, you can be satisfied applying the evaluation rules to the best of your ability to all of the functions you will encounter in your course and later, should you perchance progress further in your study, the mystery of why you are allowed to use those evaluation rules will be cleared up.