UnkleRhaukus Group Title question about notation one year ago one year ago

1. UnkleRhaukus Group Title

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. UnkleRhaukus Group Title

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

3. ash2326 Group Title

As we know that integration is independent of variable. You could use any variable of your choice. I'd go with the first one :)

4. UnkleRhaukus Group Title

a?

5. ash2326 Group Title

yes

6. UnkleRhaukus Group Title

$\int\limits_p^\infty F(a)\cdot\text da$

7. mukushla Group Title

$\int\limits_p^\infty F(x)\cdot\text dx$

8. waterineyes Group Title

$\int\limits\limits_p^\infty F(mukushla)\cdot\text d(mukushla)$

9. UnkleRhaukus Group Title

10. UnkleRhaukus Group Title

the notation is still troubling me

11. waterineyes Group Title

Where???

12. UnkleRhaukus Group Title

well i kinda had to change the question so i could answer it

13. UnkleRhaukus Group Title

\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*}

14. UnkleRhaukus Group Title

do you understand why i am getting confused/

15. waterineyes Group Title

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??

16. UnkleRhaukus Group Title

well the question in the book makes no sense

17. UnkleRhaukus Group Title

@sasogeek

18. sasogeek Group Title

certainly this is confusing, but why not use u-substitution for F(p) ?

19. UnkleRhaukus Group Title

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*}

20. sasogeek Group Title

$$let \ u=F(p)$$ $$\large \frac{du}{dp}=F'(p)$$ $$\large dp =\frac{du}{F'(p)}$$ $\int\limits_{p}^{\infty}u \ \frac{du}{F'(p)}$

21. UnkleRhaukus Group Title

im not sure why you'd do that?

22. sasogeek Group Title

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?

23. UnkleRhaukus Group Title

differential equations; laplace transforms

24. sasogeek Group Title

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 :)

25. UnkleRhaukus Group Title

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

26. UnkleRhaukus Group Title

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

27. sasogeek Group Title

you could ask satellite73 or amistre when they get online....

28. UnkleRhaukus Group Title

@amistre64

29. amistre64 Group Title

yeah, i looked at this earlier and dont have anything pertinent to add to it :/ srry