# 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 $i$ over the interval [0,N]. Let $x_{i}$ be how much of good $i$ that I consume. Let $X$ be my total consumption (the total consumption of each individual good). We can express this as $X=\int_0^N \! x_{i} \, \mathrm{d}i$.

Notice that we are integrating with respect to $i$, not $x$. I have no idea how to manipulate the above expression. Can we just take the anti-derivative with respect to $x_{i}$? How could I solve for an individual $x_{i}$? 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.