Classical Mechanics: Line Integral Example along three different paths

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Suppose you have a vector force of Fx = y, Fy = 2x. What is the work done from point A (0,0) to point B(1,1) along three different paths?

This is part of my classical mechanics course. You can find all the videos in this playlist.

Python code for 1st two paths

Parabolic path python code

00:00 - Intro
02:19 - Straight path
08:19 - Straight path in python
15:32 - Path along x y axis
18:15 - x y path in python
21:46 - parabolic path
24:58 - parabolic path in python
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Hi at 23:27 what is the math step behind this trick treating derivatives as fractions. Is that Integration by parts?substitution?

fjoensson