In "Algebraic Number Theory and Fermat's Last Theorem" by Stewart & Tall the volume is, as usually, defined as an integral. Now, exercise 6.1 (page 138) asks to show that the volume of a fundamental domain $T$ of a lattice $L \subseteq \mathbb{Z}^2$ is equal to the number of integer points lying in $T$.
I understand that by definition we have for a fundamental domain $T = \left\{ \begin{pmatrix} a \\ b \end{pmatrix}, \begin{pmatrix} c \\ d \end{pmatrix} \right\}$ with (as we known from elementary geometry) volume $vol(T) = \det(T) = ad-bc$ that
\begin{align*} vol(T) &= \int_T \ dx \ dy \\ &= \int_0^1\int_0^1 \mathbb{I}_T \ dx \ dy \\ &= ad - bc. \end{align*}
However, I do not see how to use this to count the number of integer points in $T$. Am I misunderstanding something?