- Suppose function $f(x)$ is a function giving back as output the increment of my bank account at every moment $x$ ( $x$ starting at $0$) . Suppose also ( maybe) that $f$ cannot take negative values.
The area under $C_f$ is the total amount of my bank account .
That is, if $A$ is the total amount function , $A(x) = \int_{0}^{x} f(t)dt$.
- Now , what is the ( instantaneous) rate of growth of my total amount?
By FTC, the instantaneous rate of growth of my total amount at each moment $x$ is simply $f(x)$ : $A'(x)=f(x)$.
in other words , my gain at each moment is also the ( instantaneous) rate at which my total amount is growing.
Is this correct?
Assume the total amount of your bank account changes continuously through your continuous labor and pauses. The "increment at any moment $x$" then is $0$ (otherwise you had $\infty$ in less than a minute). What can be $\ne0$ however is the $${\rm instantaneous\ increment\ intensity} \quad f(t) \qquad{\rm [dollars\ per\ minute]}\ .$$ When you have, e.g., $f(t_0)=0.5$ this means that in the tiny time interval $[t_0, \>t_0+dt]$ you earn $0.5\ dt$ dollars. Note that doubling this tiny $dt$ lets your "interval salary" double as well, which is reasonable.
Given this interpretation, and a starting total amount $a(0)$ we then have a total amount $a(T)$ at time $T>0$ given by $$a(T)=a(0)+\int_0^T f(t)\>dt\ .\tag{1}$$ When your wage (dollars per minute) is doubled for night time work then the instantaneous increment intensity $f$ is doubled during such times, and this will find its effect during the integration $(1)$.