Notation for double-integrals - partial or full differentials? When you are trying to find a volume for a function z = f(x,y), the common notation is to find:
$$\int\biggr(\int f(x,y)dx\biggr)dy$$
However, when you do this, you are actually keeping the $y$ constant on the first integral.  To me, for this to be the way it works, it seems like you should actually be using the partial differential $\partial x$ and $\partial y$.  So it seems like the notation for this should be:
$$\int\biggr(\int f(x,y)\partial_z{x}\biggr)\partial_z{y}$$
Is this an incorrect intuition?  Why or why not?

Apparently, you are correct, because when you use implicit differentation to solve partial derivatives of multivariable functions in the form $z = f(x, y)$, you will take the partial derivative of $z$ with respect to $x$, which means you will take the derivative of the $x$-terms and leave all $y$ terms as constants. It will be reversed when you take the partial derivative of $z$ with respect to $y$; you will treat all $x$-terms as constants and calculate the derivative of all the $y$-terms. Since it looks like it applies to integrals, yes, it looks like you would use $\partial x$ and $\partial y$.
