I am trying to solve a relatively simple integral on Time scales$\mathbb{T}$ in the book Dynamic equations on Time scales. It says evaluate $\int_0^t s \Delta s$ for $t \in \mathbb{T}$ for $\mathbb{T}=[0,1] \cup[2,3]$.
Here is how I proceeded, for $t \in [0,1], \mathbb{T} $ is just a subset of the real numbers so we do normal integration as in $\mathbb{R}$ to get $t^2/2$.
For $t \in [2,3]$, I think we add the integral from $[0,1]$ (i.e. $1/2$) to the integral from $2$ to $t$ which is $t^2/2 - 2$. So I get $1/2 + t^2/2 - 2 = t^2/2 - 3/2$.
Looking at the solution in the book I get the integral in $[0,1]$ correct but for $t \in [2,3]$ im wrong and the author gives the answer as $t^2/2 -1/2$ .
What is possibly wrong with my method?
It turns out I was missing the integral from 1 to 2, using Theorem 1.75 in the book mentioned above the integral from $1$ to $2$ is expressed as $$\int_1^{2}s \Delta s = \int_1^{\sigma(1)} s \Delta s = \mu(1)(1) = 1.$$
Adding $1$ to the solution in the question gives the correct answer in the book!