qmod, when you say integrals "bring out" 1/2, are you just referring to power rule? (i.e., the antiderivative of
)? Otherwise, I don't really see what you mean.
I don't really follow what you're trying to say with the factorial expressions there. Are you saying that factorials belong in the denominator because of power rule? I get that
could be seen as
, but that doesn't create a pattern with
because of that 4.
And yes, the equation for the volume of a cone fits the pattern (without the extra 4), but how on earth is a cone the 3-dimensional analog to a circle?
And I don't follow your sentence about double integration, either.
are the same thing, right? And how would you double integrate those functions of one variable? Do you mean take the antiderivative twice?
Forgive my many questions in one post, but I couldn't figure out your meaning is all.