Line Integral vs. Surface Integral: Range of t?

[nothGing]

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?

 

[HallsofIvy]

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 (x_0, y_0) to (x_1, y_1), the range of t is from whatever value of t gives x_0 and y_0 to whatever value of t gives x_1 and y_1.

 

[nothGing]

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?

 

[elibj123]

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.

 

[Redbelly98]

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.

 

[HallsofIvy]

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?

 

[nothGing]

HallsofIvy, ya.. i missunderstand already.. :p
Well, now i understand it.. thanks to elibj123,Redbelly98, HallsofIvy for helping..