Energy, power and resistance relationship

Question

This is actually a very beginner-level question but I am still not very sure about it:

Consider a circuit with a bulb:

1- Can we say that the internal resistance $\rho$ times the length $l$ of the circuit plus the resistance $R$ of the bulb is equal to the whole resistance $\Omega$ in this circuit?

2- If we wanted to calculate the energy dissipated through heat at the bulb, would it be equal to $E=\int P\ dt = \int I^2 R\ dt$, where the integration limits are $t=0$ and the time $t=T$ where the switch of the circuit is opened? And if we wanted to calculate the energy dissipated through heat over the whole circuit, that would be the same integral but with $\Omega$ instead of $R$, right?

Answer 1

It is right, if your $\rho$ is the resistance per length. Usually the resistance of the wires is so much smaller than that of the bulb, that we do not include it. If it is just a simple circuit the energy ist $U*I*t=R*I^2*t$ But if R or U or I change you are right with the integral .