High School Convention when changing integral limits

[etotheipi]

Sorry for the silly question! If we start of with the relationship $$\int_{x_{1}}^{x_{2}} F dx = KE_{2} - KE_{1}$$ and then state that at position x₁ the velocity (and hence also kinetic energy) of the particle is 0, and at x₂ its velocity is v, is it sloppy or valid to write the integral representing the work done to increase the velocity from 0 to v as $$\int_{0}^{v} F dx = KE_{2}$$ or is it necessary to change the integrand to something like the following $$\int_{0}^{v} F \frac{dx}{dv} dv = KE_{2}$$

 

[Orodruin]

It is not sloppy or valid. It is just wrong. You need to change variables to $v(x)$ to use those limits.

 

[etotheipi]

It is not sloppy or valid. It is just wrong. You need to change variables to $v(x)$ to use those limits.

Thank you, that's what I'd hoped was the case. I got confused since on this page in equation 2.1.13 the author uses limits 0 and v whilst everything still being in terms of x, which seemed a little off.

 

[RPinPA]

Yeah, that's definitely wrong. The author fixes things by changing variables at (2.1.16) so that the limits now make sense but the equations before that point are nonsense. The $v$ is just serving the role of placeholder in (2.1.13-15), "this equation is incorrect and something else goes into this position but I'm not going to bother figuring out what"