Consider a function $f(x)$ with domain $D_f=\{x\in \mathbb{R} \space|\space 0\le x\le1 \}$
Since $D_f$ is an interval of $\mathbb{R}$, it is an uncountable set and its cardinality is $\aleph_1$ under the axiomatic continuum hypothesis.
Say we want to calculate the integral $\int_{D_f}f(x)dx$. My question is: What is the cardinality of the set of all values of $x$ at which $f$ is evaluated during integration? Considering Riemann's concept of the integral as the summed up area of an infinite number of rectangles, it should be $\aleph_0$, right? (Since even though the number of rectangles goes to $\infty$, it should still be countably infinite)
The phrase "the values at which $f$ is evaluated during integration" already suffers from a bit of a misconception. The definition of the definite (Riemann) integral is a limit of finite sums. Evaluating an integral therefore doesn't really evaluate $f$ at any points directly: rather, it is the limit of a sequence of mathematical objects, each of which requires the evaluation of $f$ at finitely many points. If one wants to take the union of all these finite sets and say that they are being evaluated "during integration", then the cardinality of such evaluations is indeed $\aleph_0$. But I would recommend instead trying to change intuition away from the idea that an integral is somehow adding up infinitely many things; that's just a motivating heuristic.