How there can be an explicit coordinate dependence on the Lagrangian, if this arises from a Lagrangian density?

I have a very simple question, but strangely I cannot find any answer on the internet; maybe the answer is too simple that I don’t notice. I go straight to the point: if I define a Lagrangian from a Lagrangian density, and so from a definite integral in the coordinate space, how there can be an explicit coordinate dependence on the Lagrangian? In the picture I try to calculate the action along a fixed $s$-parametrized trajectory in the $n+1$-dimensional space in which a scalar field $phi$ is present ($y_i$ is the generical $i$-component of the $n+1$-dimensional vector $boldsymbol{y}=(boldsymbol{x},t)$ ).

begin{equation}
Sboldsymbol{=}intlimits_{tleft(s_1right)}^{tleft(s_2right)}!!!!mathrm dt,Lleft(boldsymbol{x}left(sright),tleft(sright)vphantom{tfrac{a}{b}}right)boldsymbol{doteq}!!intlimits_{tleft(s_1right)}^{tleft(s_2right)}!!!!mathrm dt!!intlimits_{mathcal M_n}!!mathrm d^n x,mathscr Lleft(phi,{partial_{y_{i}}phi},boldsymbol{x},tvphantom{tfrac{a}{b}}right)
tag{01}label{01}
end{equation}

I don’t know if it is a stupid question, but this indetermination on what actually is the nature of the quantities I’ve got in front of me, completely mess my conception up, about what I can and what I can’t do, such as when is the case to treat total and partial derivatives in the coordinate, of the lagrangian density; so thanks really in advance to anyone that will answer!