Line Integral of f(x,y,z): Exploring the Answer

In summary, the conversation discusses finding the line integral of a given function over a straight-line segment, with a given starting and ending point. The correct solution is 3√(14), but the speaker initially got a negative result due to using the wrong starting point. The speaker also asks about the significance of direction in the integral and determining the interval for the integral.
  • #1
dwn
165
2

Homework Statement



Find the line integral of f(x,y,z) = x+y+z over the straight-line segment from (1,2,3) to (0,-1,1).

Homework Equations



∫ f(x,y,z)ds = ∫ f(g(t), h(t), k(t)) |v(t)| dt

The Attempt at a Solution



I arrived at the correct solution, but I'd like some clarity on the result.

The final answer to this is 3√(14) or -3√(14) depending on which point you choose as your parametric equation.
x = -t
y = -3t-1
z = -2t+1
From using the point (0,-1,1) and (-1,-3,-2) as my direction vector.

What I would like to understand is the meaning of the positive and negative result. Does it matter?
It just seems to me that my result should have been positive since I am moving from a lower position to a higher position, no?BTW, QUICK SHOUTOUT TO PF --- THE NEW SITE IS AMAZING! GREAT JOB ON THE NEW LAYOUT.
 
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  • #2
dwn said:

Homework Statement



Find the line integral of f(x,y,z) = x+y+z over the straight-line segment from (1,2,3) to (0,-1,1).

Homework Equations



∫ f(x,y,z)ds = ∫ f(g(t), h(t), k(t)) |v(t)| dt

The Attempt at a Solution



I arrived at the correct solution, but I'd like some clarity on the result.
Which is the correct solution? You show 2 different values below.
The final answer to this is 3√(14) or -3√(14) depending on which point you choose as your parametric equation.
x = -t
y = -3t-1
z = -2t+1
From using the point (0,-1,1) and (-1,-3,-2) as my direction vector.

How did you arrive at this parameterization for the line segment specified? For example, if t = 0, does your parameterization return the (x,y,z) of the first point on the line segment?

What I would like to understand is the meaning of the positive and negative result. Does it matter?

Yes, it matters. Some line integrals are path independent. Is this one?

It just seems to me that my result should have been positive since I am moving from a lower position to a higher position, no?

How did you arrive at this conclusion? The OP states that the line integral is to be taken over the line segment from (1,2,3) to (0,-1,1), not the other way around.
 
  • #3
Correct Answer: 3√(14) but I got the negative term because I used the wrong point as my starting position.Why does direction of the integral matter though? Because the value of the integral will remain the same. I see that they are asking us to go from a specified point and not the other, but is it really necessary?
 
Last edited:
  • #4
Is ##\int_{a}^{b} f(x) dx = \int_{b}^{a} f(x) dx ## ?
 
  • #5
No it is not.

I actually have a question about a and b though. How do we determine the interval? I still don't quite understand that.
 
Last edited:

1. What is a line integral of f(x,y,z)?

A line integral of f(x,y,z) is a mathematical concept used in multivariable calculus to calculate the total value of a function along a specific path or curve in three-dimensional space.

2. How is a line integral of f(x,y,z) calculated?

A line integral of f(x,y,z) is calculated by breaking down the path into small segments, approximating the function value at each segment, and adding them together to get an overall approximation of the total value. As the number of segments approaches infinity, the approximation becomes more accurate.

3. What is the significance of the direction of the path in a line integral of f(x,y,z)?

The direction of the path in a line integral of f(x,y,z) is important because the value of the integral can change depending on the direction of the path. If the path is reversed, the resulting integral will have the opposite sign.

4. What are some real-world applications of line integrals of f(x,y,z)?

Line integrals of f(x,y,z) have many applications in physics, engineering, and other sciences. They can be used to calculate work done by a force along a specific path, electric and magnetic fields along a wire, and fluid flow rates in a pipe, among others.

5. Are there any alternative methods for calculating line integrals of f(x,y,z)?

Yes, there are alternative methods for calculating line integrals of f(x,y,z) such as Green's theorem, Stokes' theorem, and the Divergence theorem. These theorems relate line integrals to surface integrals and volume integrals, allowing for a different approach to solving them.

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