Finding the arc length of a parabola.

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burt

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I was asked to show that to find the length of an arc of the parabola \(y=x^2\) one needs to determine the area under the hyperbola \(y^2-4x^2=1\).


How can I go about this?
 
burt said:
I was asked to show that to find the length of an arc of the parabola \(y=x^2\)

one needs to determine the area under the hyperbola \(y^2-4x^2=1\).
Click to expand...
That statement does not make sense without specified boundaries.
 
"...one needs to..." -- This is simply false. The fact that the two calculations result in the same NUMERICAL expression is of little consequence. One is a linear measure. The other is a measure of area. Just not the same thing.

Show your two calculation. Let's see you demonstrate it.
 
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Do you know the formula (using calculus) for calculating the arc length of a curve?
 
How do you suppose to show where the two curves overlap, if in fact they overlap?
 
Subhotosh Khan said:
That statement does not make sense without specified boundaries.
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I agree. I think you need to assume corresponding boundaries.

The arc length of \(\displaystyle y=f(x)\) from \(\displaystyle x=a\) to \(\displaystyle x=b\) is given by:

Arc length \(\displaystyle = \int_a^b \sqrt{1+(f '(x))^2}\, dx\)

Burt, try this with \(\displaystyle f(x) = x^2\) and I think you will see what the question is getting at.
 
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