Integral with an index

Question

I have an economic problem where it is claimed that the production of final goods is given by the following integral
$$
Y=int_{i=0}^{i=1}x_idi
$$
where $x_i$ is the production of firms in an intermediate sector. In the intermediate sector there are infinite firms producing the intermediate product $x_i$.
I will appreciate if anyone can help me to understand three things:

How do you understand the integral over a subindex?
How do you solve this integral?
How do you compute $frac{partial Y}{partial x_i}$?

Thanks in advance!

orthxx · Answer

From what I know about such models, $x_i$ should be considered as a continuous function of $i$: $x(i)$ that maps $[0,1]$ to $mathbb{R}$. This means that you have infinitely many firms producing some goods. The integral would be just
$$Y=int_0^1 x(i)di.$$
However, I do not think that the integral $frac{partial Y}{partial x_i}$ makes sense as a variation of $x$ on a zero-measure set does not change the value of the integral.

Integral with an index

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