How to evaluate if a function of two or more variables is odd with respect to one of those variables?

Question

I am solving this question which asks a function of x and y to be integrated over a volume of a particular cylinder.

But all of this working can be skipped by realising the original function is an odd function with respect to x, so the integral is equal to zero.

So my question is, is how can you tell if a function is odd if the function is a function of more than one variable?

Please see the working out below the question.

Is the first approach I take correct? Evaluating f((-x),y) and seeing if it is equal to -f(x,y) ?