I am working through some basic probability stuff and have a question regarding functions of multiple variables. If I have two random variables $X,Y$ which have some joint probability distribution $P_{XY}(x,y)$ I can obtain expected value of some function $f(x,y)$ by integrating across both variables: $$ E[f(x,y)] = \int \int f(x,y) P_{XY}(x,y) \ dx \ dy $$
we can also obtain the expected value of a function of a single variable by following the workings here $$ E[g(x)] = \int g(x) P_{X}(x) \ dx $$
If the integration across a function of a single value has meaning I am wondering how we would interpret the integration of a function of multiple variables across a single value $$ \int f(x,y) P_X(x)dx $$
and wether this expression has any meaning?
The last expression gives you the expectation of $f(X,y)$ where $X$ denotes the random variable and $y$ is fixed.
So defining $Z_y:=f(X,y)$ it gives the expectation of random variable $Z_y$.