# Line Integral question

by mathsciguy
Tags: integral, line
 P: 132 Suppose I have a vector V and I want to compute for the line integral from point (1,1,0) to point (2,2,0) and I take the path of the least distance (one that traces the identity function). The line integral is of the form: $$\int _a ^b \vec{V} \cdot d\vec{l}$$ Where: $$x=y, \ d\vec{l} =dx \hat{x} + dx \hat{y}$$ Thus the integral can be computed purely in terms of x (can also be y), which looks something like this: $$\int _a ^b V(x)dx$$ What I don't exactly understand is why is it okay to use the limits like this: $$\int _1 ^2 V(x)dx$$ Why can we use the limits from 1 to 2 if we express the line integral in terms purely of x. I have a very vague idea of why it is, but I'd rather take it from people who actually know this to explain this to me. Thanks.