In Terence Tao's paper Differential Forms and Integration, he mentions that there are $3$ distinct notions of integration when discussing functions $f: \Bbb R \to \Bbb R$
- Indefinite Integrals: $\int f(x)\ dx$
- Unsigned Definite Integrals: $\int_{[a,b]} f(x)\ dx$
- Signed Definite Integrals: $\int_a^b f(x)\ dx$
I know well the definitions of indefinite integral — $\int f(x)\ dx = F(x) \iff F'(x) = f(x)$ — and the signed definite integral — via the Darboux or Riemann sum definitions. But I've never heard of an unsigned definite integral and I can't find a rigorous definition of it.
What is the definition of the unsigned definite integral?
Best Answer
Given a set $S$ in a measure space and a measure $dx$, you can consider the integral $$ \int_S f dx$$ of an integrable function $f$. For instance, we might look at $$ \int_{[0,1]} 1 dx = 1.$$
One might pronounce this as an integral of the constant function $1$ over the interval from $0$ to $1$.
There is no way to associate a sign to the specification of the set. The set has no orientation, to borrow a term from integration on manifolds.
We recognize this as being the same as $$ \int_0^1 1 dx = 1.$$ But this latter notation is signed, as evidenced by the natural pronunciation as the integral of the constant function $1$ from $0$ to $1$. With notation, it also makes sense to talk about $$ \int_1^0 1 dx = -1.$$
In this sense, this latter integral is signed.
More generally, there are signed integrals over any differentiable manifold. There is unsigned differentiation over any measure space.