Calculating Length of Line cf: A Math Problem Explained

[ewr]

In the problem attached, why is the integral become just the length of line cf?

 

[Nugatory]

View attachment 75602
In the problem attached, why is the integral become just the length of line cf?

That result pretty much follows from the definition of a line integral. If you're not familiar with that concept, google will find you some pretty decent explanations.

 

[jtbell]

Note that in part (a) the integral becomes just the length of line if, that is, d.
$$V_f - V_i = -\int^f_i {\vec E \cdot d\vec s}
= -\int^f_i {E(\cos 0^\circ)ds} = -E \int^f_i {ds} = -Ed$$