TITLE: What does the integral of position with respect to time mean?
QUESTION [22 upvotes]: The integral of acceleration with respect to time is velocity.
The integral of velocity with respect to time is position.
What is the integral of position with respect to time, and what does it mean?
Please explain so that your answer is understandable by someone who took calculus I.

REPLY [2 votes]: Suppose there's a lever that you can move from $0$ to $1$. Letting $f(t)$ be the position of that lever over time as you move it, you can think of the lever's velocity $f'$, acceleration $f''$, etc. Now imagine that lever controls a floodgate. The floodgate is closed at the $0$ position and opens as you move the lever towards $1$. The integral $\int_0^t f(x)\,\mathrm{d}x$ measures the accumulation of water that has spilled from the floodgate over time. If you leave the lever at some fixed position, the water will be flowing out at a constant rate. The absement in this situation measures the accumulated water, equal to $\int_0^t f(x)\,\mathrm{d}x$, and is a measure of the sustained position of the lever.