I know how to do it, but it's ridiculously long.
How do I do this the fast way, assuming there is one.
I found the area of all and removed the negative areas to get the distance.
And correctly got A.
Now, if you'd been given the start position (call it a) and end position (call it b), then there would be an easier way. It would be (1/14)(b-a). (By the fundamental theorem of calculus.)
Edit: You could however eliminate B immediately, since there's no way a denominator of 14 could get simplified to 16. You could eliminate E by eyeballing it. But I don't see easy ways to eliminate C and D.
So going from point A to B, the area is 0 because the positive and negative contributions are equal.
Then from B to C it's -18.
Then from C to the place where it crosses the x-axis, it's -6, making overall total of -24. Then from that point to D it's 24, making overall total 0.
Then from D to E it's 2 * (average of 8 and 12) = 20. So the answer is 20/14 = 10/7.