Find Length of Curve Between Two X Values

In summary, finding the length of a curve between two x values is possible by using the element of length formula and integrating it. This can be done in both two and three dimensions, using Cartesian coordinates or converting from other coordinate systems. It is not necessary to apologize for any language mistakes, as your English is excellent.
  • #1
soandos
166
0
is there a way to find the length of a curve between two x values?
if so, what is it.
thanks
 
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  • #2
Yes since the element of length is (assuming two dimensions)

[tex]ds=\sqrt{dx^2+dy^2}[/tex]

If you integrate you get the length of the curve [itex]s[/itex] from [itex][a,b][/itex]

[tex]s=\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt=\int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}dx[/tex]
 
  • #3
In three dimensions, write x, y, z as functions of parameter t and do the same:
[tex]\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dt}\right)^2+ \left(\frac{dz}{dt}\right)^2}dt[/tex]
 
  • #4
I am sorry for my broken english.

How can one find the length of a curve, if the coordinate system is not rectangular (for instance, it is spherical)?

Please if not inconvenient to you, point out my mistakes in my english.
 
  • #5
By using exactly the same formulas but converting from the other coordinate system.

For example, suppose a path is given in spherical coordinates by [itex]\rho= 1[/itex], [itex]\phi= \pi/3[/itex], [itex]\theta= t[/itex], with parameter t. In Cartesian coordinates that is [itex]x= \rho cos(\theta) sin(\phi)= (\sqrt{3}/2)cos(t)[/itex], [itex]y= \rho sin(\theta)sin(\phi)= (\sqrt{3}/2)sin(t)[/itex], [itex]z= \rho cos(t)= cos(t)[/itex].

And my only criticism is that you should stop apologizing for your English. It is excellent. Far better than my (put whatever language you wish in here!).
 

1. What is the purpose of finding the length of a curve between two x values?

The length of a curve between two x values is a mathematical calculation that is used to determine the distance along a curved line between two specific points. This can be useful in many fields, including physics, engineering, and mathematics.

2. How is the length of a curve between two x values calculated?

The length of a curve between two x values is calculated using a mathematical concept known as integration. This involves breaking the curve into smaller segments and using a formula to find the length of each segment. The lengths are then added together to find the total length of the curve between the two x values.

3. What factors can affect the accuracy of the length of a curve between two x values?

There are several factors that can affect the accuracy of the length of a curve calculation. These include the precision of the measurement points, the complexity of the curve, and any errors in the mathematical calculations used to find the length.

4. Can the length of a curve between two x values be negative?

No, the length of a curve between two x values cannot be negative. The length of a curve is always a positive value, as it represents the distance between two points on a curve.

5. How is the length of a curve between two x values used in real-world applications?

The length of a curve between two x values has many practical applications, such as calculating the distance traveled by a moving object, determining the curvature of a road or track, and finding the surface area of a curved object. It is also used in engineering and design to create smooth and accurate curves in various structures and systems.

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