Integral with respect to a measure over a subset

Question

The constuction of the integral of a function $f:X \rightarrow \mathbb R $ with respect to a measrue $\mu$ i.e. $$ \int _X f \mathrm d \mu.$$ is well known. I'm wondering what if I want to integrate $f$ over a subset $A \subset X$. Does it make any difference if we define $$ \int _A f \mathrm d\mu:=\int _X f 1_A \mathrm d \mu $$ or $$ \int _A f \mathrm d \mu:=\int _A f|_A \mathrm d \mu$$?