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nameVoid
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How Does Arc Length Calculation Relate to Surface Problems?
- Thread starter nameVoid
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In summary, The problem at hand is finding the arc length along a curve between points A and B. The integrand for this calculation is sqrt(1 + (y')^2), which simplifies to a perfect square when rewritten as \sqrt{1 + (\frac{x^2}{4} - \frac{1}{x^2})^2}. This makes it easy to integrate and obtain the arc length. The question about surfaces does not seem relevant to this specific problem.
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Mark44
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What, are we supposed to guess what the problem is from the title on the thread? Presumably you want to find the arc length along the curve between points A and B.
What does this have to do with surfaces, though?
For the arc length, the integrand is sqrt(1 + (y')^2), which can be written as
[tex]\sqrt{1 + (\frac{x^2}{4} - \frac{1}{x^2})^2}[/tex]
[tex]=\sqrt{1 + \frac{x^4}{16} -1/2 + \frac{1}{x^4}}[/tex]
The last three terms under the radical are a perfect square. When you add the first term, you'll still have a perfect square, which makes it easy to take the square root, which means you'll have an easy function to integrate.
What does this have to do with surfaces, though?
For the arc length, the integrand is sqrt(1 + (y')^2), which can be written as
[tex]\sqrt{1 + (\frac{x^2}{4} - \frac{1}{x^2})^2}[/tex]
[tex]=\sqrt{1 + \frac{x^4}{16} -1/2 + \frac{1}{x^4}}[/tex]
The last three terms under the radical are a perfect square. When you add the first term, you'll still have a perfect square, which makes it easy to take the square root, which means you'll have an easy function to integrate.
Related to How Does Arc Length Calculation Relate to Surface Problems?
1. What is arc length?
Arc length is the distance along the curved line of an arc from one endpoint to the other. It is measured in the same units as the radius of the arc.
2. How is arc length calculated?
Arc length can be calculated using the formula L = rθ, where L is the arc length, r is the radius of the arc, and θ is the central angle in radians.
3. What is the difference between arc length and arc measure?
Arc length is the actual distance along the arc, while arc measure is the size of the central angle subtended by the arc. Arc measure is measured in degrees or radians, while arc length is measured in units of length.
4. Can arc length be negative?
No, arc length cannot be negative. It is always a positive value, as it represents a distance.
5. How is arc length related to the circumference of a circle?
Arc length is a fraction of the circumference of a circle. The fraction is determined by the central angle subtended by the arc. For example, if the central angle is 90 degrees, the arc length will be one-fourth of the circumference of the circle.
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