Electrical Potential of Semicircle Insulating Wire

[stod]

Homework Statement

A thin insulating rod is bent into a semicircular wire of radius a, with a total charge of Q distributed uniformly along the rod. Calculate the potential at the center of the curvature if the potential is assumed to be zero at infinity.

Homework Equations

V=ʃ (kdq/r)

The Attempt at a Solution

I integrated the equation from zero to Q, and my final answer was kQ/a, which was the same as the book's solution manual. My problem is, the solution manual did it a completely different and longer way, and I'm just wondering if my method is not valid (because the book does it the same way as me on some similar examples).

the book made dq=(lambda)dl, where lambda=Q/pi*a, and dl=a*dθ.
Then, after moving stuff around, the final integral looked like this:

integral(k*Q/pi*a)dθ, and they integrated from zero to pi.


I've spent a few hours trying to figure out why you can just integrate dq from zero to Q in some instances, while in other instances that seem very similar (like this) you have to make dq equal something. Or, did the book do this as a matter of preference, and it is okay to integrate dq from zero to Q in this instance? Please help clear up this extreme fogginess for me. Thank you in advance to anyone who can help me.

 

[rl.bhat]

In the case of potential you can use your method. Because the net potential is the scalar addition of potential due to dq. But if you want to find the electric field at the center, you cannot use this method.