Line integral

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what is the different between line integral and surface integral?
If we parameterize curve by x=t , y=t , what is the range of t ? Is it 0<= t <=1? why?
 

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  • #2
HallsofIvy
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what is the different between line integral and surface integral?
A "line integral" is along a one-dimensional line and a "surface integral" is over a two dimensional surface.

If we parameterize curve by x=t , y=t , what is the range of t ? Is it 0<= t <=1? why?
There are no bounds. if you are talking about a portion of a curve, say from [itex](x_0, y_0)[/itex] to [itex](x_1, y_1)[/itex], the range of t is from whatever value of t gives [itex]x_0[/itex] and [itex]y_0[/itex] to whatever value of t gives [itex]x_1[/itex] and [itex]y_1[/itex].
 
  • #3
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Erm.. ok..
Let say, integral ( x^2 + y + z)ds , the line segment of curve is from (0,0,0) to (1,2,1).
We parameterize curve C by x=t , y=2t , z=t.
As you said, t should be between 0 and 2.
but according to my reference book, 0<=t<=1.
Why?
 
  • #4
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HallsofIvy said that the range of t is such that your parametrization will agree with your end points. (Of course, you first need to check that your parametrization does describe a line, in this case it does).

With your paramterization, its 0<=t<=1. But I could choose:

x=t/2, y=t, z=t/2

Which still describe a line with the same orientation, instead now t must range from 0 to 2 to give the desired segment.
 
  • #5
Redbelly98
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Erm.. ok..
Let say, integral ( x^2 + y + z)ds , the line segment of curve is from (0,0,0) to (1,2,1).
We parameterize curve C by x=t , y=2t , z=t.
As you said, t should be between 0 and 2.
but according to my reference book, 0<=t<=1.
Why?
When t=1, x,y,z is the final endpoint. Try plugging t=1 into the forumulas for x,y,z to convince yourself of this.
 
  • #6
HallsofIvy
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Erm.. ok..
Let say, integral ( x^2 + y + z)ds , the line segment of curve is from (0,0,0) to (1,2,1).
We parameterize curve C by x=t , y=2t , z=t.
As you said, t should be between 0 and 2.
No, I did NOT say that! I said " if you are talking about a portion of a curve, say from (x0, y0) to (x0, y0) , the range of t is from whatever value of t gives x0 and y0 to whatever value of t gives x1 and y1" . Since this is in three dimensions, we need to include z0 and z1 also.

We must have x(t)= t= 0, y(t)= 2t= 0, and z(t)= t= 0. Obviously t= 0 satisfies all of those.

We must also have x(t)= t= 1, y(t)= 2t= 2, and z(t)= t= 1. Obviously t= 1, NOT t= 2, satisfies all of those.

but according to my reference book, 0<=t<=1.
Why?
 
  • #7
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HallsofIvy, ya.. i missunderstand already.. :p
Well, now i understand it.. thanks to elibj123,Redbelly98, HallsofIvy for helping..
 

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