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Rotated frame of reference

1. The problem statement, all variables and given/known data
Last week I https://www.physicsforums.com/showthread.php?t=483819" which I believe didn't get answered because my description of the problem was a lot messier than it needs to be. Hopefully today, I will not make the same mistake.

In physics books, when calculating the electric field due to a finite line of charge, the problem is usually setup as shown in the top half of my figure. The point of interest / evaluation is placed on the Y axis, and the two endpoints of the line are placed right on the x axis.

What I would like to know is how to set up my integral to solve for this basic problem, if my line weren't parallel with the X axis. I know I could rotate everything by some angle theta (the angle between the line and the x-axis), do my calculations, then rotate everything back into place. But I would much rather discover the correct way to arrange my integrals to solve for this sort of problem directly, rather than altering the reference frame.


2. Relevant equations
Please see attached diagram below.


3. The attempt at a solution
[PLAIN]http://authman.net/problem.jpg [Broken]


Thanks in advance for any direction that is bestowed..
 
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