Convention when changing integral limits

In summary, the conversation discusses the validity of using the integral $$\int_{0}^{v} F dx = KE_{2}$$ to represent the work done in increasing a particle's velocity from 0 to v, when the initial and final positions are x1 and x2 respectively. The experts conclude that this is not valid and that the integral needs to be changed to $$\int_{0}^{v} F \frac{dx}{dv} dv = KE_{2}$$ by changing variables to ##v(x)##. They also note that the author of the original source may have used incorrect equations before fixing them later on.
  • #1
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 $$\int_{0}^{v} F \frac{dx}{dv} dv = KE_{2}$$
 
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  • #2
It is not sloppy or valid. It is just wrong. You need to change variables to ##v(x)## to use those limits.
 
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  • #3
Orodruin said:
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.
 
  • #4
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"
 
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Related to Convention when changing integral limits

1. What is the purpose of changing integral limits?

The purpose of changing integral limits is to simplify the integral and make it easier to solve. By changing the limits, we can often transform the integral into a more familiar form that can be evaluated using known techniques.

2. How do you change integral limits?

To change integral limits, we use a substitution or transformation method. This involves replacing the original variable in the integral with a new variable, and then adjusting the limits accordingly. The new limits should correspond to the new variable.

3. When should you change integral limits?

You should change integral limits when the original limits make it difficult to evaluate the integral or when the integral can be simplified by using a substitution or transformation. This is often the case when dealing with complicated functions or when trying to evaluate improper integrals.

4. What are some common conventions when changing integral limits?

Some common conventions when changing integral limits include using the same variable for the new limits, using the inverse function for trigonometric substitutions, and using the derivative of the substitution variable for integration by parts.

5. Can changing integral limits affect the value of the integral?

Yes, changing integral limits can affect the value of the integral. This is because the limits determine the range over which the function is being integrated. By changing the limits, we are essentially evaluating the integral over a different interval, which can result in a different value.

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