Quick Linear Integration Question

In summary, the conversation is about breaking an integrand into two integrals, F_y over the y component of dh and F_x over the x component, for linear integration over the hypotenuse of a right triangle with equal, undefined Δx Δy sides. The problem involves an undefined function F and the question is whether it can be solved by parametrizing F_y and F_x. After some discussion, it is concluded that the solution involves integrating over dh and replacing x and y with F_x and F_y, respectively. The problem is then identified as a "textbook type" problem and further discussion ensues about the best approach to solving it.
  • #1
sriracha
30
0
If my integrand is:

[F_y (dy/dh) + F_x (dx/dh)] dh

Can I break this into two integrals, F_y over the y component of dh and F_x over the x component:

[F_y] (dh)_y + [F_x] (dh)_x

This is for linear integration over the hypotenuse of a right triangle with equal, undefined Δx Δy sides. F is also undefined.
 
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  • #2
I guess this can't be right because F_x can depend on y and vice-versa. I will post the question and the work I have done. Please note this is not a homework assignment. I have looked at other similar problems, but none where F is undefined and this is what is giving me problems.
 
  • #3
The problem is attached and I have uploaded my work here: http://i39.tinypic.com/kbd4s4.jpg

(I wanted to put it in high resolution and the file was too big for PF).
 

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  • #4
By the way I arbitrarily chose the midpoint of the hypotenuse as my (x,y,z) although I realize now it would make things a bit clearer if I had chosen (x[itex]_{0}[/itex],y[itex]_{0}[/itex], z[itex]_{0}[/itex]). I'm pretty certain this should have no bearing on the work I did on the left.

I am wondering If I can go ahead and integrate once I have the problem solved to the point where I have [itex]\frac{1}{\Delta h}[/itex][itex]\int[/itex][itex]^{\Delta h}_{0}[/itex] F[itex]_{y}[/itex] - F[itex]_{x}[/itex] dh ?
 
  • #5
That doesn't seem right either because then I'm left with F_y - F_x and I to integrate over h I think I would need to parametrize F_y and F_x. Even if I do, I am left with F_x and F_y of (deltay - h / deltah). I need a respective F_y (x,y,z) *deltay - F_x (x,y,z) * delta x and guess I have no idea how to get there.
 
  • #6
Yay me! Where I found x = deltay - h/deltah and y = h/deltah I realized I could replace the x and ys with F_x and y with F_y then integrate over dh.
 
  • #7
Also realized this is a "textbook type" problem so you can move it if you want.
 
  • #8
http://www.infoocean.info/avatar1.jpg I guess this can't be right because F_x can depend on y and vice-versa.
 
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  • #9
bwood01 said:
http://www.infoocean.info/avatar1.jpg I guess this can't be right because F_x can depend on y and vice-versa.

That's what I said.
 
Last edited by a moderator:

1. What is quick linear integration?

Quick linear integration is a mathematical process used to find the area under a straight line on a graph. It is a form of numerical integration and is commonly used in scientific and engineering fields to approximate the value of an integral.

2. How is quick linear integration different from other integration methods?

Unlike other integration methods, quick linear integration only works for calculating the area under a straight line. It is a simplified method that does not require advanced mathematical techniques, making it a quick and easy way to approximate integrals.

3. When is quick linear integration typically used?

Quick linear integration is commonly used when a more accurate integration method is not necessary and a quick estimation of the integral is needed. It is also useful when the function being integrated is a straight line, as it simplifies the calculation process.

4. What are the limitations of quick linear integration?

Quick linear integration can only be used for approximating integrals of straight lines. It is not suitable for more complex functions, and the accuracy of the approximation decreases as the number of intervals used in the calculation increases.

5. How do I perform quick linear integration?

To perform quick linear integration, you will need to divide the area under the line into smaller intervals and calculate the area of each interval using the formula for the area of a rectangle (base x height). Then, add up the areas of all the intervals to get an approximation of the integral.

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