Thread: Integrate with respect to an index

1. Integrate with respect to an index

Hello all,

This problem comes up quite frequently in economics (and I am completely stuck...)

I have the choice to spread my consumption between a continuum of goods (indexed by $\displaystyle i$ over the interval [0,N]. Let $\displaystyle x_{i}$ be how much of good $\displaystyle i$ that I consume. Let $\displaystyle X$ be my total consumption (the total consumption of each individual good). We can express this as $\displaystyle X=\int_0^N \! x_{i} \, \mathrm{d}i$.

Notice that we are integrating with respect to $\displaystyle i$, not $\displaystyle x$. I have no idea how to manipulate the above expression. Can we just take the anti-derivative with respect to $\displaystyle x_{i}$? How could I solve for an individual $\displaystyle x_{i}$? Any advice (no matter how little), would be greatly appreciated.

Cheers,

Nick

2. I don't see any danger in reading x_i as x(i) analogous to f(x)... so x is your vertical/output axis (usually y or f(x)), i your horizontal, input axis (instead of x), then proceed (and integrate) as normal.