When Calculating the Inner Product, Why Do We (Seemingly) Always Integrate From $0$ to $1$? I've always seen the inner product $\langle f(x), g(x)\rangle$ written as
$$\langle f(x), g(x)\rangle = \int_0^1 f(x)g(x)w(x) \ dx$$
where $w(x)$ is a weight function.
But why do we always integrate from $0$ to $1$? And why not other values?
If we do integrate over other values, why is integrating over the domain from $0$ to $1$ so common?
I would greatly appreciate it if people could please take the time to clarify this.
 A: It's probably just a function of the material. The eigenfunctions are typically just orthogonal between $a$ and $b$ i.e.
$$ \int_{a}^{b} \phi_{n}(x)\phi_{m}(x) \sigma(x) dx =0  , \lambda_{n} \neq \lambda_{m} $$
typically in PDEs you start off with $0$ to $L$  i.e 
$$ \int_{0}^{L} \phi_{n}(x) \phi_{m}(x) \sigma(x) dx  $$
and $\sigma(x) =1$ , $ \lambda_{n} = \left( \frac{n \pi }{L} \right)^{2} $ then $  \phi_{n}(x) = \sin(\frac{ n \pi x}{L})$ for instance
when we have 
$$ \frac{d^{2}\phi}{dx^{2}} + \lambda \phi =0  \\ \phi(0) = 0 \\ \phi(L) = 0$$ 
A: Consider the following two questions, in order:


*

*What is the simplest number you can think of, especially one you might use as a starting point?

*What is the next simplest number you can think of, especially one you might use as an ending point?


I posit that by far the most common answers to these questions are $0$ and $1$ respectively. Therefore, this is the example that gets used the most.
Taking $0 \to 1$ as your "reference" path has an additional advantage in that if you want to convert to another path $a \to b$, it's extremely easy to recreate the affine formula for it:
$$ t \mapsto a (1-t) + b t $$
