Theoretical/ Dimensional Analysis Question

Question

Position is the integral of velocity. However, position and velocity have different dimensions. How is this difference consistent with the conclusion that the integration sign is dimensionless?

Answer 1

the sign itself is dimensionless, but simply speaking, what you are doing is $$ \int_a^b f(x) dx = \lim_{n \to \infty} \sum_{k=1}^n f(x_k) \delta x_k, $$ where $f$ is velocity and $\delta x_k$ has spatial dimension...