How is the Total Electric Field Calculated for a Ring?

[djMan]

Hi I am pretty confused on how my book is doing the calculus of electric fields. Basically I don't understand how their equation makes any sense (The integral equations on the second page). How does the indefinite integral become a definite integral? Is this a true equality or is it supposed to just represent a concept?

 

[Bashkir]

It is because your limits of integration will change depending on the surface you are dealing with. In this case we are dealing with a ring, then we can consider each little bit of the ring having an infinitesimal charge, dq. We can find a linear charge density that that says there is so much charge for so much little bit of ring and call that lambda as shown in your pictures. To find the total charge then we have to sum up all of the little bits of charge over the total ring. Since this is a ring, we are only concerned with the circumference and we integrate with our upper bound being the circumference of the ring.

This obviously changes based on what you are integrating. An indefinite integral is purely algebraic, it was just describing the action done. The indefinite integral becomes definite when we are considering the analytic (geometric) properties of our object.