# Integrating a differential and notation of limits of integration

I am studying Thermodynamics and in class I came up with integrating the differential of internal energy:$$\int^2_1dU=U_2-U_1$$ But I don't understand why someone couldn't just write $$\int^2_1dU=U|^2_1=2-1=1$$ So how can we know if the integral mean the difference between two different function states?

• $U_2$ needs not be equal to $2$, and likewise, $U_1$ needs not be equal to $1$. Commented Feb 10, 2020 at 22:32
• There is an abuse of notation for the integral. The limits should be $U_1$ and $U_2$. What your integral means that it's integrating the internal energy between states "1" and "2", not between values $1$ and $2$ Commented Feb 10, 2020 at 22:34
• @Andrei Thanks for the answer. Our proffesor did that abuse yesterday. Commented Feb 10, 2020 at 22:34
• You would evaluate it incorrectly. It’s an abuse of notation. Commented Feb 10, 2020 at 22:36

It's an abuse of notation. If $$U$$ is a function of $$t$$, say, then $$dU=U'(t)\,dt$$, the initial and final states are $$U_1=U(1)$$, $$U_2=U(2)$$. So what the integral is trying to mean is $$\int_1^2\,dU=\int_1^2\,U'(t)\,dt=\left.U(t)\right|_1^2=U(2)-U(1).$$