Finding the arc length of a parabola.

[burt]

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?

 

[Deleted member 4993]

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 $y^2-4x^2=1$.

That statement does not make sense without specified boundaries.

 

[tkhunny]

"...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.

 

[Deleted member 4993]

Makes sense. I was assuming they meant the overlapping portion.

Which overlapping portion?

 

[Harry_the_cat]

Do you know the formula (using calculus) for calculating the arc length of a curve?

 

[Steven G]

How do you suppose to show where the two curves overlap, if in fact they overlap?

 

[Harry_the_cat]

That statement does not make sense without specified boundaries.

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.