.Line Integral in the First Quadrant

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In summary: Thanks for the correction.In summary, the conversation discusses the use of Green's theorem to evaluate a line integral along a closed path in the first quadrant, formed by the x-axis, the line x=1, and the curve y=x^3. Some individuals suggest using Green's theorem, while others suggest parametrizing the curves and using FTC. It is also mentioned that polynomials are analytic and that the domain should be described by specific values for x and y. Finally, a summary of the process for integrating along the path is provided.
  • #1
kidia
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Question:
Starting from anyone of the three corners of the path described here under,evaluate the line integral [tex]\int[/tex][tex]c[/tex] (2xy[tex]^3[/tex])dx + (4x[tex]^2[/tex]y[tex]^2[/tex])dy where C is the closed path forming the boundary of region in the first quadrant enclosed by x-axis,the line x=1 and the curve y=x[tex]^3[/tex]
 
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  • #2
Well, what have you got so far?
 
  • #3
kidia said:
Question:
Starting from anyone of the three corners of the path described here under,evaluate the line integral [tex]\int[/tex][tex]c[/tex] (2xy[tex]^3[/tex])dx + (4x[tex]^2[/tex]y[tex]^2[/tex])dy where C is the closed path forming the boundary of region in the first quadrant enclosed by x-axis,the line x=1 and the curve y=x[tex]^3[/tex]

Any time you have a nice closed path with the line integral in that form, try and remember to use Green's Theorem:

[tex]\oint_C Mdx+Ndy=\iint_R(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y})dA[/tex]

Just plug it in right?
 
  • #4
Not so fast saltydog - I don't think he is allowed to use Green or Stokes for now.

just parametrize the curves, find derivative of that curve and then you can find individual line integrals of int(F dot dr) for all segments
 
  • #5
It's easier to compute the double integral than the line integral...So i vote for Green as well.

Daniel.
 
  • #6
I might be embarrassing myself here but isn't there some sort of requirement that the function be analytic before you use Green's Theorem? And not all polynomials are analytic.
 
  • #7
i don't think green has much hypotheses? maybe smoothness? try proving it and see what you need.

it follows from fubini (repeated integration) plus ftc.

of course all polynomials are analytic, but i am asuming you meant that you thought the form had to be closed, which is not necessary.
 
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  • #8
That domain should be described by x running from 0 to 1 and y from 0 to x^{3}...

Daniel.
 
  • #9
snoble said:
I might be embarrassing myself here but isn't there some sort of requirement that the function be analytic before you use Green's Theorem? And not all polynomials are analytic.

All the polynomials I know are analytic!

(And you don't need the function to be analytic for Green's theorem- "continuously differentiable" is enough.)

The only question is whether kidia has had "Green's theorem" in Calculus class yet.

It's not that hard to actually integrate along the path:
1) from (0,0) to (1, 0) along the x- axis: take x= t, y= 0 so that dx= dt, dy= 0. Since y= 0 along that path, the integral on it is 0.

2) from (1,0) to (1,1) along the line x= 1:take x= 1, y= t so that dx= 0, dy= dt. integrate
4t2dt from 0 to 1.

3) from (1,1) to (0,0) along the line y= x: take x= t, y= t so that dx= dt, dy= dt. integrate from t= 1 down to t= 0.
 
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  • #10
I knew I was setting myself up for embarresment.
 

Related to .Line Integral in the First Quadrant

1. What is a Line Integral in the First Quadrant?

A Line Integral in the First Quadrant is a type of mathematical calculation used to determine the total value of a function over a specific curve or path in the first quadrant of a coordinate plane.

2. How is a Line Integral in the First Quadrant calculated?

A Line Integral in the First Quadrant is calculated by dividing the curve into small segments and finding the area under each segment using the function's value at that point. The sum of these areas gives the total value of the Line Integral.

3. What is the significance of calculating a Line Integral in the First Quadrant?

Calculating a Line Integral in the First Quadrant can help in finding the work done by a force in a specific direction, calculating the mass or charge distribution along a curve, and finding the path of a particle in a vector field.

4. What are the differences between a Line Integral in the First Quadrant and a Regular Integral?

The main difference between a Line Integral in the First Quadrant and a Regular Integral is that the former is calculated over a specific curve or path, while the latter is calculated over an interval on the x-axis. Also, the Line Integral takes into account the direction of the curve, whereas the Regular Integral does not.

5. How is a Line Integral in the First Quadrant applied in real life?

A Line Integral in the First Quadrant has various applications in physics, engineering, and economics. It is used to calculate the work done by a force in a specific direction, the flow of fluid through a curved pipe, and the circulation of a vector field. It is also used in economics to determine the cost of production along a specific path.

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