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#1 2010-10-06 01:07:47

calccrypto
Member
Registered: 2010-03-06
Posts: 96

Line Integral with changing variables

How do you change variables properly to get the line integral of any f(x,y,z) on any curve C modeled by r(t)??? Im so lost!


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#2 2010-10-06 06:06:24

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Line Integral with changing variables

Hi calccrypto;

I am the worst in the world with line integrals. How about posting a specific problem and maybe we can get through it together.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2010-10-06 06:55:02

calccrypto
Member
Registered: 2010-03-06
Posts: 96

Re: Line Integral with changing variables

Hi bobbym! I got this question:

C modeled by  r(t) = (t, 3t^2, 6t^3)
and
f(x,y,z) = x y z^2
compute the line integral of f along C as modeled by r(t)

so far, i have only gotten r'(t) = (1, 6t, 18t^2) and f(t) = 18t^6. I dont know how to continue. all i know is the answer is 864/5

Last edited by calccrypto (2010-10-06 07:05:16)


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#4 2010-10-06 07:42:23

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Line Integral with changing variables

Hi;

How did you get 18t^6 for f(r(t))?

Did they give you some range for t?

Like maybe 0 <= t <= 1 ?

Take a look here for some general info that might help.

http://calculus.mindsay.com/section_172 … egrals.mws


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#5 2010-10-06 09:15:50

calccrypto
Member
Registered: 2010-03-06
Posts: 96

Re: Line Integral with changing variables

i just replaced the components of r for (x,y,z) and multiplied.

and yep. I forgot to put in the limits, which are 0<= t <= 1 like you guessed


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#6 2010-10-06 09:24:06

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Line Integral with changing variables

Hi;

If I guessed the interval then you did not do f(r(t)) right. Want to try it again?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#7 2010-10-06 09:34:43

calccrypto
Member
Registered: 2010-03-06
Posts: 96

Re: Line Integral with changing variables

i dont know how to continue at all


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#8 2010-10-06 10:19:48

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Line Integral with changing variables

Hi;

This is what I did:

Please check the work as I have not done one of these in years and my notes on them are sketchy.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#9 2010-10-06 12:45:38

calccrypto
Member
Registered: 2010-03-06
Posts: 96

Re: Line Integral with changing variables

oh.. thats what you meant in that previous post. wow. i fail. shoot me

Last edited by calccrypto (2010-10-06 12:46:45)


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#10 2010-10-06 14:06:32

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Line Integral with changing variables

Hi cal;

No shooting allowed here. Had my brother check it, he said this is a book problem that he remembers but he can not remember where.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#11 2010-10-06 23:22:45

calccrypto
Member
Registered: 2010-03-06
Posts: 96

Re: Line Integral with changing variables

No wonder! I forgot to square the z component.


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#12 2010-10-06 23:49:30

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Line Integral with changing variables

Hi;

Use this formula when you are given a line integral where everything has already been parametized like r(t).

It is all covered in the link I posted. This link here will provide more and probably what you will be doing soon.

http://ocw.mit.edu/courses/mathematics/ … s/embed30/


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#13 2010-10-11 13:15:04

calccrypto
Member
Registered: 2010-03-06
Posts: 96

Re: Line Integral with changing variables

what happens if f returns a vector?


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#14 2010-10-11 14:12:13

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Line Integral with changing variables

Hi calccrypto;

I do not see how that can happen from the above formula.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#15 2010-10-11 23:15:54

calccrypto
Member
Registered: 2010-03-06
Posts: 96

Re: Line Integral with changing variables

i meant a generic question. instead of xyz^2, say the output of f was something like (xy,yz,xz) or something. then what?


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#16 2010-10-12 07:11:36

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Line Integral with changing variables

Hi calccrypto;

You mean when you substitute x,y,z, in f(x,y,z)?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#17 2010-10-12 15:32:09

calccrypto
Member
Registered: 2010-03-06
Posts: 96

Re: Line Integral with changing variables

no i meant instead of

f(x,y,x) = scalar value like xyz

the equation is

f(x,y,z) = vector, like (xy,yz,xz)

but i think i got it


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