$$
[\text{size of boundary}] \times [\text{rate of motion of boundary}] = [\text{rate of change of size of bounded region}]
$$
This differs from the fundamental theorem of calculus in that it does not mention antiderivatives and you can present it on the first day of a beginning calculus course, with concrete examples. It immediately appeals to the intuition of even relatively non-mathematically inclined students. I have a forthcoming paper on the use of this proposition in teaching.
My question is: Is there some standard name for this statement?
And does it appear in the literature somewhere other than in my paper?
Added in a later edit: Several answers say this is a special case of something else (just as I did above). But in each case the "something else" is far too abstruse to present on the first day of the freshman calculus course. So let's add a different question: What should this simple statement be called?
Best Answer
I believe what you wrote down is the differential form of the co-area formula. (Unfortunately, the version on Wikipedia is too "advanced" or "modern" so it is hard to see immediately that when you integrate both sides of your equation in time, you get precisely the co-area formula.)