# Integral with an index

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:

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

Mathematics Asked on November 12, 2021

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.

Answered by orthxx on November 12, 2021

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