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.
$\endgroup$ 2What is the definition of the unsigned definite integral?
2 Answers
$\begingroup$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.
$\endgroup$ $\begingroup$He kind of defines the unsigned definite integral in the second paragraph — or at least relates it to the more familiar signed definite integral. Noting that he's restricted attention to the real numbers, we take $a, b \in \mathbf{R}$ such that $a \leq b$, and a function $f: \mathbf{R} \to \mathbf{R}$. His equation (2) says \begin{equation} \tag{2} \int_a^b f(x)\, dx = -\int_b^a f(x)\, dx = \int_{[a,b]} f(x)\, dx. \end{equation} We can use this as the definition of the last quantity. I guess if you want to be more complete, you can also define the integral when $a > b$. In this case, the interval $[a, b]$ is conventionally the empty set, and an integral over the empty set is conventionally $0$.
But it's important to note that we can conceptually distinguish between the Darboux or Riemann sums implicit in the first two terms of Eq. (2) and the "integration over a set" implicit in the last. Even though they have the same value, they are conceptually distinct. And so, when we generalize to other spaces with different structures on those spaces, we expect these two types of integral to generalize in different ways.
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