Work calculation in a thermodynamics cycle process

Question

To calculate work in a cyclic process (say Carnot cycle) we find area under p-v curve,or use net heat transfer= net work done. Why won't we simply add work calculated in each process of a cycle,like in Carnot: W1 +W2 +w3 + w4

Answer 1

Why won't we simply add work calculated in each process of a cycle,like in Carnot: W1 + W2 + W3 + W4

We can (But you need to be careful in calling energy of each process as work, $W_1$ & $W_3$ as you call them are heats added and taken from the system not mechanical work).

You already answered your question, the net work of the Carnot cycle is the area enclosed under the $p-v$ curve.

$$ W_{net} = \oint_{\text{cycle}} P\ dV = \oint_{\text{cycle}} T\ dS$$

You can calculate the area under each process in $p-v$ diagram (some will be negative) and the sum will be the net work $W_{net} = Q_{in} - Q_{out}$ (enclosed in white in the following figure, but the axis are temperature and entropy, but you get the idea).