UnkleRhaukus
  • UnkleRhaukus
question about notation
Mathematics
  • Stacey Warren - Expert brainly.com
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chestercat
  • chestercat
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UnkleRhaukus
  • UnkleRhaukus
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
UnkleRhaukus
  • UnkleRhaukus
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
ash2326
  • ash2326
As we know that integration is independent of variable. You could use any variable of your choice. I'd go with the first one :)

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UnkleRhaukus
  • UnkleRhaukus
a?
ash2326
  • ash2326
yes
UnkleRhaukus
  • UnkleRhaukus
\[\int\limits_p^\infty F(a)\cdot\text da\]
anonymous
  • anonymous
\[\int\limits_p^\infty F(x)\cdot\text dx\]
anonymous
  • anonymous
\[\int\limits\limits_p^\infty F(mukushla)\cdot\text d(mukushla)\]
UnkleRhaukus
  • UnkleRhaukus
1 Attachment
UnkleRhaukus
  • UnkleRhaukus
the notation is still troubling me
anonymous
  • anonymous
Where???
UnkleRhaukus
  • UnkleRhaukus
well i kinda had to change the question so i could answer it
UnkleRhaukus
  • UnkleRhaukus
\[\begin{align*} \int\limits_p^\infty F(q)\cdot\text dq &=\int\limits_p^\infty\int\limits_0^\infty f(x)e^{-qx}\cdot\text dx\cdot\text dq \\&=\int\limits_0^\infty f(x)\int\limits_p^\infty e^{-qx}\cdot\text dq\cdot\text dx \\&=\int\limits_0^\infty f(x)\left.\frac{e^{-qx}}{-x}\right|_p^\infty \cdot\text dx \\&=\int\limits_0^\infty \frac{f(x)}xe^{-px}\cdot\text dx \\&=\mathcal L\left\{\frac {f(x) }x \right\} \end{align*}\]
UnkleRhaukus
  • UnkleRhaukus
do you understand why i am getting confused/
anonymous
  • anonymous
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??
UnkleRhaukus
  • UnkleRhaukus
well the question in the book makes no sense
UnkleRhaukus
  • UnkleRhaukus
@sasogeek
sasogeek
  • sasogeek
certainly this is confusing, but why not use u-substitution for F(p) ?
UnkleRhaukus
  • UnkleRhaukus
like this? \[\begin{align*} \int\limits_p^\infty F(p)\cdot\text dp\\ \text{let }u=p\\ \text du=\text dp\\ \\p=p\rightarrow u=u\\ p=\infty\rightarrow u=\infty\\ \\&=\int\limits_u^\infty F(u)\cdot\text du\\ \end{align*}\]
sasogeek
  • sasogeek
\(let \ u=F(p)\) \(\large \frac{du}{dp}=F'(p)\) \(\large dp =\frac{du}{F'(p)} \) \[\int\limits_{p}^{\infty}u \ \frac{du}{F'(p)}\]
UnkleRhaukus
  • UnkleRhaukus
im not sure why you'd do that?
sasogeek
  • sasogeek
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?
UnkleRhaukus
  • UnkleRhaukus
differential equations; laplace transforms
sasogeek
  • sasogeek
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 :)
UnkleRhaukus
  • UnkleRhaukus
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
UnkleRhaukus
  • UnkleRhaukus
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
sasogeek
  • sasogeek
you could ask satellite73 or amistre when they get online....
UnkleRhaukus
  • UnkleRhaukus
@amistre64
amistre64
  • amistre64
yeah, i looked at this earlier and dont have anything pertinent to add to it :/ srry

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