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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?

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    $\begingroup$ $U_2$ needs not be equal to $2$, and likewise, $U_1$ needs not be equal to $1$. $\endgroup$ Commented Feb 10, 2020 at 22:32
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    $\begingroup$ 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 $2$ $\endgroup$
    – Andrei
    Commented Feb 10, 2020 at 22:34
  • $\begingroup$ @Andrei Thanks for the answer. Our proffesor did that abuse yesterday. $\endgroup$
    – user599310
    Commented Feb 10, 2020 at 22:34
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    $\begingroup$ You would evaluate it incorrectly. It’s an abuse of notation. $\endgroup$ Commented Feb 10, 2020 at 22:36

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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 $$ \int_1^2\,dU=\int_1^2\,U'(t)\,dt=\left.U(t)\right|_1^2=U(2)-U(1). $$

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