A quick question regarding integral notation.

Question

Say a particle is traveling along the parameterized curve $\left(X,Y\right)$, where $X$ and $Y$ are functions of time. I want to find the distance it has traveled as a function of time. The solution would be something like this: $$\int_0^t\sqrt{\dot{X}^2+\dot{Y}^2}\,dt$$ This integral is with respect to $t$, and the integral is on the interval $[0,t]$. Is it correct to have $t$ in both places? I would like to avoid using another variable.

Answer 1

It's not correct. You're forcing $t$ to have an out-of-body experience. It has to go to $0$ and travel back to itself. There's no getting out of a new variable.