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## Main Question or Discussion Point

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:

[tex] \int _a ^b \vec{V} \cdot d\vec{l} [/tex]

Where:

[tex] x=y, \ d\vec{l} =dx \hat{x} + dx \hat{y} [/tex]

Thus the integral can be computed purely in terms of x (can also be y), which looks something like this:

[tex] \int _a ^b V(x)dx [/tex]

What I don't exactly understand is why is it okay to use the limits like this:

[tex] \int _1 ^2 V(x)dx [/tex]

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.

The line integral is of the form:

[tex] \int _a ^b \vec{V} \cdot d\vec{l} [/tex]

Where:

[tex] x=y, \ d\vec{l} =dx \hat{x} + dx \hat{y} [/tex]

Thus the integral can be computed purely in terms of x (can also be y), which looks something like this:

[tex] \int _a ^b V(x)dx [/tex]

What I don't exactly understand is why is it okay to use the limits like this:

[tex] \int _1 ^2 V(x)dx [/tex]

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.