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How Does Arc Length Calculation Relate to Surface Problems?
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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.