Why do the heat absorbed and work done depend on the path of a thermodynamic process?

In thermodynamics, the reason heat ($Q$) and work ($W$) depend on the path of a process is that they are not state functions.

Unlike internal energy ($U$), which is determined solely by the current state of a system (such as its temperature, pressure, and volume), $Q$ and $W$ represent energy in transit across the system's boundary. A system can move from an initial state to a final state through various processes—each involving different combinations of energy exchange—while resulting in the same net change in internal energy ($\Delta U=Q-W$).12

Mathematically, this is expressed by the fact that the infinitesimal changes in heat ($d^{\prime }Q$) and work ($d^{\prime }W$) are not exact differentials. Because their values depend on the specific sequence of states (the path) taken to reach the final condition, their definite integrals are path-dependent. For example, in gas expansion, the work done is represented by the area under the curve on a $P$-$V$ diagram; different paths between the same two states yield different areas, and thus different amounts of work.1324