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UnkleRhaukus Group Title

question about notation

  • 2 years ago
  • 2 years ago

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  1. UnkleRhaukus Group Title
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    in my book there is a question involving a term like this \[\int\limits_p^\infty F(p)\cdot\text dp \], but this is a confusing choice of notation because the variable of integration is one one of limits of integration

    • 2 years ago
  2. UnkleRhaukus Group Title
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    as far as i know the convention is to change to \[\int\limits_p^\infty F(p')\cdot\text dp'\] but i dont like this because it looks like the prime of p

    • 2 years ago
  3. ash2326 Group Title
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    As we know that integration is independent of variable. You could use any variable of your choice. I'd go with the first one :)

    • 2 years ago
  4. UnkleRhaukus Group Title
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    a?

    • 2 years ago
  5. ash2326 Group Title
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    yes

    • 2 years ago
  6. UnkleRhaukus Group Title
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    \[\int\limits_p^\infty F(a)\cdot\text da\]

    • 2 years ago
  7. mukushla Group Title
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    \[\int\limits_p^\infty F(x)\cdot\text dx\]

    • 2 years ago
  8. waterineyes Group Title
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    \[\int\limits\limits_p^\infty F(mukushla)\cdot\text d(mukushla)\]

    • 2 years ago
  9. UnkleRhaukus Group Title
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    • 2 years ago
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  10. UnkleRhaukus Group Title
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    the notation is still troubling me

    • 2 years ago
  11. waterineyes Group Title
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    Where???

    • 2 years ago
  12. UnkleRhaukus Group Title
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    well i kinda had to change the question so i could answer it

    • 2 years ago
  13. UnkleRhaukus Group Title
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    \[\begin{align*} \int\limits_p^\infty F(q)\cdot\text dq &=\int\limits_p^\infty\int\limits_0^\infty f(x)e^{-qx}\cdot\text dx\cdot\text dq \\&=\int\limits_0^\infty f(x)\int\limits_p^\infty e^{-qx}\cdot\text dq\cdot\text dx \\&=\int\limits_0^\infty f(x)\left.\frac{e^{-qx}}{-x}\right|_p^\infty \cdot\text dx \\&=\int\limits_0^\infty \frac{f(x)}xe^{-px}\cdot\text dx \\&=\mathcal L\left\{\frac {f(x) }x \right\} \end{align*}\]

    • 2 years ago
  14. UnkleRhaukus Group Title
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    do you understand why i am getting confused/

    • 2 years ago
  15. waterineyes Group Title
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    The things you are doing are going above my head, but I am not understanding why are you getting confused, where are you not feeling comfortable??

    • 2 years ago
  16. UnkleRhaukus Group Title
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    well the question in the book makes no sense

    • 2 years ago
  17. UnkleRhaukus Group Title
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    @sasogeek

    • 2 years ago
  18. sasogeek Group Title
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    certainly this is confusing, but why not use u-substitution for F(p) ?

    • 2 years ago
  19. UnkleRhaukus Group Title
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    like this? \[\begin{align*} \int\limits_p^\infty F(p)\cdot\text dp\\ \text{let }u=p\\ \text du=\text dp\\ \\p=p\rightarrow u=u\\ p=\infty\rightarrow u=\infty\\ \\&=\int\limits_u^\infty F(u)\cdot\text du\\ \end{align*}\]

    • 2 years ago
  20. sasogeek Group Title
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    \(let \ u=F(p)\) \(\large \frac{du}{dp}=F'(p)\) \(\large dp =\frac{du}{F'(p)} \) \[\int\limits_{p}^{\infty}u \ \frac{du}{F'(p)}\]

    • 2 years ago
  21. UnkleRhaukus Group Title
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    im not sure why you'd do that?

    • 2 years ago
  22. sasogeek Group Title
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    well having the function variable being the same as one of the limits... i'm not quite sure how you'd go about solving it but thats the first thing that comes to my mind. besides, i'm not so good at calculus, but that's what i'd do. it may or may not be right, but it's worth a shot :/. I'm pretty certain this isn't calculus 1... right?

    • 2 years ago
  23. UnkleRhaukus Group Title
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    differential equations; laplace transforms

    • 2 years ago
  24. sasogeek Group Title
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    there we go, that's beyond me lol, but all things being equal, thats what i'd do. sorry i wasn't of much help lol :/ but i'll stick around to learn :)

    • 2 years ago
  25. UnkleRhaukus Group Title
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    but the problem im having is not the content of the question, i think i have answered that satisfactorily , my issus is the choice of notation used in the question

    • 2 years ago
  26. UnkleRhaukus Group Title
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    i dont think the question makes sense as written \[....=\int\limits_p^\infty F(p)\text dp\] it dosent make sense to have the variable as a limit of integration

    • 2 years ago
  27. sasogeek Group Title
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    you could ask satellite73 or amistre when they get online....

    • 2 years ago
  28. UnkleRhaukus Group Title
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    @amistre64

    • 2 years ago
  29. amistre64 Group Title
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    yeah, i looked at this earlier and dont have anything pertinent to add to it :/ srry

    • 2 years ago
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