Minimizing Area Under Curve: COV

In summary, the conversation discusses finding the shape of a curve that minimizes the area underneath it between two given points. It is determined that this is an Euler-Lagrange problem, with specific constraints and equations of motion that need to be solved to find the curve.
  • #1
opsb
27
0
So, If you've got two points and a given length of curve to 'hang' between them, what shape is the curve which minimises the area underneath it? For a curve which is almost the same length as the distance between the points, this would be a catenary, I think (a la famous hanging chain problem), but for longer curves it would be different. Any ideas?
 
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  • #2
So you want to minimize the integral

[tex]I=\int_a^b f(x)dx[/tex]

with the constraints

[tex]f(a)=A[/tex]

[tex]f(b)=B[/tex]

[tex]L=\int_a^b\sqrt{1+f'^2}dx[/tex]

It's an Euler-Lagrange problem. The lagrangian is

[tex]\mathscr{L}=f+\lambda\sqrt{1+f'^2}[/tex]

so the equations of motion are

[tex]\frac{d}{dx}\frac{\lambda f'}{\sqrt{1+f'^2}}=1[/tex]

in other words

[tex]\frac{\lambda f'}{\sqrt{1+f'^2}}=cx+d[/tex]

You have to find c, d and lambda using the constraints above. Then you have to solve for f '. Finally you integrate (this is the hard part) to find f.
 

FAQ: Minimizing Area Under Curve: COV

1. What is COV and why is it important in minimizing area under curve?

COV stands for coefficient of variation, which is a measure of the variability of a set of data. It is important in minimizing area under curve because it helps us understand the spread of the data points and how much they deviate from the mean. This is crucial in determining the most accurate and efficient way to minimize the area under the curve.

2. How does minimizing COV contribute to minimizing area under curve?

Minimizing COV helps to reduce the variability in the data, which in turn makes it easier to accurately predict and minimize the area under the curve. This is because a lower COV indicates a more uniform and consistent data set, making it easier to determine the optimal curve.

3. What methods can be used to minimize COV?

There are several methods that can be used to minimize COV, including data cleaning and filtering to remove outliers, standardizing the data to a common scale, and using statistical techniques such as regression analysis to identify and eliminate sources of variability.

4. How can COV be calculated and interpreted?

COV can be calculated by dividing the standard deviation of the data by its mean. A lower COV indicates a more consistent and predictable data set, while a higher COV suggests a greater degree of variability and unpredictability in the data.

5. Are there any limitations to using COV in minimizing area under curve?

While COV is a useful measure in minimizing area under curve, it does have some limitations. For example, it assumes that the data follows a normal distribution, and may not accurately capture the variability in skewed or non-normal data sets. It is important to use COV in conjunction with other statistical techniques to get a more comprehensive understanding of the data and minimize the area under the curve effectively.

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