Any ideas for #1? Not really sure, but I think it's related to some type of mean theorem.

 

I think it is probably a straight-forward plou and chug. Could be wrong. What is the expression for the mean of the function in terms of the parameters given? That will probably help.

 

I think one also makes a substitution for the function within the triple integral - where the substitution is the Taylor's series expansion of the function in three dimensions. Then one decides which terms within the integration vanish and which ones don't - a matter which is presumably intuitive for mathematicians.

 

I've attached a copy of a paper which outlines the method of derivation ...

 

Very nice, tnk, thank you... I'm busy working on something else right now, but will read this tonight.