Seeking help understanding definition of Riemann sum

So, my textbook gives the definition of a Riemann sum in the screenshot below. I understand most of the definition, but I'm lost on $c_i$ and what the i'th sub-interval is. Previously I was finding the definite integral by:

$$\lim_{n \to \infty} \sum_{i=1}^n f(m_i)\cdot \Delta x$$

or

$$\lim_{n \to \infty} \sum_{i=1}^n f(M_i) \cdot \Delta x$$

where $f(m_i$) are the minimum endpoints and $f(M_i$) are the maximum endpoints. But now my textbook is showing me examples where it's best to use $f(c_i$) instead of the endpoint

Question

Seeking help understanding definition of Riemann sum

So, my textbook gives the definition of a Riemann sum in the screenshot below. I understand most of the definition, but I'm lost on $c_i$ and what the i'th sub-interval is. Previously I was finding the definite integral by:

$$\lim_{n \to \infty} \sum_{i=1}^n f(m_i)\cdot \Delta x$$

or 

$$\lim_{n \to \infty} \sum_{i=1}^n f(M_i) \cdot \Delta x$$

where $f(m_i$) are the minimum endpoints and $f(M_i$) are the maximum endpoints. But now my textbook is showing me examples where it's best to use $f(c_i$) instead of the endpoint

anonymous · Answer

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anonymous · Answer

(continued)

instead of the endpoints. If anybody can offer an explanation (perhaps a visual example) I'd greatly appreciate it. I'd also quite like any resources on this, as I couldn't find anything about $(c_i)$. I looked on google and khanacademy but didn't find anything.

P.S. Sorry for the long post(s), I just don't know what info is relevant or not. Thanks in advance!