Notation and terminology of definite integrals I am currently undertaking Calculus and have stumbled upon an issue regarding notation and terminology, that I can not seem to understand.
"Given the definition of the definite integral:
If f is a function on the interval [a,b] and we divide the interval into n subintervals of equal width where $x_i^*$ is the $i^{th}$ interval then the definite interval of f from a to b is:
$\int^b_a f(x)dx $ = $\lim\limits_{n\to\infty} \sum_{i=1}^n f(x_i^*)\Delta(x)$"
Now, my understanding of the definite integral is that it is the limit of the Riemann sum. However, I was doing a question in my lecture and noticed that my professor had a question that asked us to "evaluate the following definite integral:
$\int^2_0 (x^2+3x) dx$"
Calculating it is not an issue but the lead me to question what we actually consider the definite integral. Is it the limit of the Riemann sum as I mentioned or is it the notation that we use, namely $\int^b_a f(x) dx$?
Is it perhaps a clumsy way of wording a question or am I missing something?
I look forward to your responses!
Thank you in advance
 A: You should think of
$$
\int^2_0 (x^2+3x) dx
\tag1$$
as
$$
\int_a^b f(x)\;dx
$$
where $a=0, b=2$ and $f$ is the function described by $f(x) = x^2+3x$.
Yes, $(1)$ is the limit of Riemann sums, where of course $f(x_i^* )$ is
$$
(x_i^*)^2+3x_i^*
$$
A: 
...[this] lead me to question [of] what we actually consider [to be] the definite integral. Is it the limit of the Riemann sum as I mentioned or is it the notation that we use, namely $\displaystyle \int_{a}^{b}f(x) dx$?

I believe your confusion stems from how you are reading $$\displaystyle \int^b_a f(x)dx  = \lim\limits_{n\to\infty} \sum_{i=1}^n f(x_i^*)\Delta(x).$$ In particular, it might be better to look at the right hand side of the equals sign $$\lim\limits_{n\to\infty} \sum_{i=1}^n f(x_i^*)\Delta(x)\tag{$1$}$$ as the definition of  $$\displaystyle \int_{a}^{b}f(x) dx\tag{$2$}.$$ We use the notation $(2)$ in practice to calculate definite integrals instead of working with $(1)$ as it is much cleaner and convenient. Better yet, we have a set of rules on how to calculate definite integrals using $(2)$. In that respect if someone asked me to evaluate the definite integral $$\displaystyle \int_{0}^{2}(x^{2} + 3x)dx$$ I would use $(2)$ to do so.
