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Work check on some integrals

  • Thread starter WWCY
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  • #1
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Homework Statement



Hi everyone, I'd appreciate it if someone could help look through my working and check if it makes sense!

I have the following integral:
$$P(t) = \int_{-\infty}^{a} \int_{-\infty}^{-\infty} f_p(p) \ f_{x} (x - pt/m) \ dp \ dx$$
I want to find an expression for ##\frac{dP(t)}{dt}##, so I first "bring in" the integral over ##x##.
$$P(t) = \int_{-\infty}^{-\infty} f_p(p) \Big [ \int_{-\infty}^{a} f_x (x - pt/m) \ dx \Big ] \ dp$$
Making the substitution of ##u = x - pt/m##
$$\frac{d}{dt} \int_{-\infty}^{a} f_x (x - pt/m) \ dx = \frac{d}{dt}\int_{-\infty}^{a - pt/m} f_x (u) \ du = -\frac{p}{m} f_x(a - pt/m)$$
and finally,
$$\frac{dP}{dt} = -\frac{1}{m} \int_{-\infty}^{\infty} f_p (p) . p . f_x (a - pt/m) \ dp$$
Is this right? Many thanks in advance.

Homework Equations




The Attempt at a Solution

 

Answers and Replies

  • #2
jambaugh
Science Advisor
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You can also differentiate directly. In general:
[tex]\frac{d}{dt} \int^{b(t)}_{a(t)} f(x,t)dx = f(b(t),t) b'(t) - f(a(t),t) a'(t) + \int^{a}_b \frac{\partial}{\partial t} f(x,t)dt[/tex]
In your case there is (originally) no ##t## dependency in the limits so:

Let's see...
[tex]\frac{d}{dt}P(t) = \frac{d}{dt} \int_{-\infty}^{a} \int_{-\infty}^{\infty} f_p(p) \ f_{x} (x - pt/m) \ dp \ dx =[/tex]
[tex] = \int_{-\infty}^{a} \int_{-\infty}^{\infty} f_p(p) \ f'_{x} (x - pt/m)\frac{-p}{m} \ dp \ dx[/tex]
changing the order of integration:
[tex] = \int_{-\infty}^{\infty}\frac{-p}{m}f_p(p) \int_{-\infty}^{a}\ f'_{x} (x - pt/m) \ dx \ dp[/tex]
and applying the FTC in its other form (using u substitution for the indefinite integral ##du = dx## and returning to ##x## for the evaluation):
[tex]= \int_{-\infty}^{\infty}\frac{-p}{m}f_p(p) \ f_{x} (a - pt/m) dp[/tex]

so we agree.

I had some concern with your change of variable which has a ##t## dependency before executing the derivative which is why I worked it out in this alternative fashion. That's something I should meditate on for a bit to be sure if and why it is okay.
 
Last edited:
  • #3
475
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Thanks a lot for your time!
 

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