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What does taking the integral of x(t) yield? I'm very inquisitive and far ahead of my school math, but i have yet to understand what the result is.

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Do you mean $ ∫ x(t) dt $ , where $ x(t)$ denotes time varying position ? – Souvik Dey Nov 9 '12 at 5:41
Honestly -- it is meaningless. In physics, of course. Try Googling it. Have you considered a much more interesting question -- what is the derivative of acceleration? It actually has a physical meaning. – glebovg Nov 9 '12 at 5:44

It yields many many many many many many things.

One of them is the average value. If you want to know how big $x(t)$ is on average for $t$ between $t=a$ and $t=b$, find the integral of $x(t)$ from $t=a$ to $t=b$, and divide by $b-a$.

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