- #1

Loren Booda

- 3,125

- 4

What is the measure of the sin(x) wave for x=0 to 2∏?

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

In summary, the measure of the sin(x) wave from x=0 to 2∏ is given by the integral \int^{2\pi}_0\sqrt{\cos(x)^2+1} dx. This is equivalent to the length of the graph of y= f(x) from x= a to x= b, which is represented by the integral \int_{x=a}^b \sqrt{1+ f'(x)^2}dx. However, this cannot be solved using elementary functions and requires methods such as numerical integration, Simpson's rule, or approximations.

- #1

Loren Booda

- 3,125

- 4

What is the measure of the sin(x) wave for x=0 to 2∏?

Mathematics news on Phys.org

- #2

disregardthat

Science Advisor

- 1,866

- 34

It's [tex]\int^{2\pi}_0\sqrt{\cos(x)^2+1} dx[/tex]

- #3

Loren Booda

- 3,125

- 4

That's what I got. Would one need a table of integrals to determine its numerical value?

- #4

HallsofIvy

Science Advisor

Homework Helper

- 42,988

- 975

The length of the graph of y= f(x), from x= a to x= b, is given by

[tex]\int_{x=a}^b \sqrt{1+ f'(x)^2}dx[/tex]

With y= f(x)= sin(x), f'(x)= cos(x) so that becomes

[tex]\int_{x=0}^{2\pi} \sqrt{1+ cos^2(x)}dx[/tex]

However, that looks to me like a version of an elliptical integral which cannot be done in terms of elementary functions.

Hey, no fair posting while I'm typing!

- #5

- 15,464

- 690

Yep. It's 4√2E(1/2), where E(x) is the complete elliptical integral of the second kind.HallsofIvy said:However, that looks to me like a version of an elliptical integral which cannot be done in terms of elementary functions.

- #6

mathwonk

Science Advisor

Homework Helper

- 11,774

- 2,041

this is no worse than finding the area under the curve from 0 to 1. i.e. both are approximations.(nobody knows what cos(1) is.)

When we say "determine the length of a curve," we are referring to finding the total distance along the curve from one point to another. This is commonly done in mathematics and physics to understand the shape and properties of a curve.

The length of a curve can be calculated using a mathematical method called integration. This involves breaking the curve into infinitesimally small segments and adding them together to find the total length. For the curve sin(x), we use a specific integration formula to determine its length.

The curve sin(x) is a simple and well-known function that follows a smooth, continuous path. This makes it an ideal example for calculating the length of a curve as it is easy to visualize and understand. Moreover, the integration formula for this curve is relatively straightforward, making it a good introductory example for students.

No, the length of a curve cannot be determined without using integration. This is because the concept of length is inherently linked to the concept of distance, which requires integration to be accurately calculated. However, there are some approximation methods that can be used to estimate the length of a curve without using integration.

The concept of determining the length of a curve is used in many fields, including engineering, physics, and computer graphics. For example, it is used to calculate the length of a path taken by a moving object, the shape of a rollercoaster track, or the trajectory of a projectile. In computer graphics, the length of a curve is used to create smooth and realistic animations.

- Replies
- 2

- Views
- 1K

- Replies
- 1

- Views
- 938

- Replies
- 1

- Views
- 867

- Replies
- 5

- Views
- 1K

- Replies
- 2

- Views
- 937

- Replies
- 11

- Views
- 1K

- Replies
- 2

- Views
- 1K

- Replies
- 3

- Views
- 1K

- Replies
- 1

- Views
- 872

- Replies
- 2

- Views
- 1K

Share: