Definition of $\underline{\int_{a}^{b}} f(x) \,dx$ and $\overline{\int_{a}^{b}} f(x) \,dx$ unclear

Question

I don't quite understand the definition of $\underline{\int_{a}^{b}} f(x) \,dx$ and $\overline{\int_{a}^{b}} f(x) \,dx$. Should I interpret $\underline{\int_{a}^{b}} f(x) \,dx$ as the largest area that is part of a lower sum and $\overline{\int_{a}^{b}} f(x) \,dx$ as the smallest area that is part of the upper sum, but then this no longer seems logical to me (*): \begin{align} \underline{S}(f,V)\leq \underline{\int_{a}^{b}} f(x) \,dx \leq \overline{\int_{a}^{b}} f(x) \,dx \leq \overline{S}(f,V) \end{align} where $V=\{a=x_0<x_1<...<x_n=b\}$ a partition of the interval $[a,b]$ and $a,b\in \mathbb{R}$.
The largest area that is part of a lower sum is not greater than or equal to the entire lower sum, right? Can someone help me with the definitions of $\underline{\int_{a}^{b}} f(x) \,dx$ and $\overline{\int_{a}^{b}} f(x) \,dx$?

$\underline{S}(f,V)$ is the lower sum, $\overline{S}(f,V)$ is the upper sum, $\underline{\int_{a}^{b}} f(x) \,dx$ is the lower integral and $\overline{\int_{a}^{b}} f(x) \,dx$ is the upper integral.