Generalizing Riemann Sums

A function $f(x)$ is differentiable for all $x$ over the interval $[a, b]$ where $a, b \in \mathbb{Z}$

Approximating $\displaystyle\int_{a}^{b}f(x)dx$ using a left Riemann sum with 1 unit subintervals is equivalent to evaluating which of the following?

A. $\displaystyle\int_{a}^{b}\lfloor f(x) \rfloor dx$

B. $\displaystyle\int_{a}^{b}f(\lfloor x \rfloor)dx$

C. $\displaystyle\int_{a}^{b}\lceil f(x) \rceil dx$

D. $\displaystyle\int_{a}^{b}f(\lceil x \rceil)dx$

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Notation:

$\lfloor \cdot \rfloor$ denotes the floor function .

$\lceil \cdot \rceil$ denotes the ceiling function .