I am struggling with this concept of the parametrization being path independent. It is true I asked a similar question before but I don't believe I did an adequate explanation.
Start with a line integral over the path $x = y^2$. and we use the parametrization $x = t$ and $y = t^2$ over $0 < t < 1$. ( The specific line integral is not given because it is not needed for the question.) Now we can integrate with respect to t and come up with an area that represents work! I am OK with this much.
Now let us change the parametrization to $x = sin(t)$ and $y = sin^2(t)$ over $0 < t < \frac{\pi}{2}$. The work will be the same and the power will be the same.
My question: what exactly is happening when we parameterize? Is there another function in the background that follows the same path in the same time? I would like to get an intuitive sense of what parametrization does but is not affected by time since the power is the same. Thank you .
This is an animation of the area below $y=x^2$ accumulated for $x = t$ and $x = \sin \frac{\pi t}{2}$ parametrizations as $t \in [0,1]$ varies. The integral computed is always this area, which is clearly independent of how fast the curve is traversed.