What Are Fat Curves in Parametrics and How Do They Relate to Lengths?

[hangainlover]

Homework Statement

The curves with equations x" + y" = 1, n = 4, 6, 8, , are called fat circles. Graph the curves with n = 2, 4, 6, 8, and 10 to see why. Set up an integral for the length S(2k), the arc of the fat circle with n=2k. Without attempting to evaluate the integral, state the value of lim S(2k) as x approaches infinity.

Homework Equations

Formula for the lengths of curves.
L= intergal of square root ((dx/dt)^2)/(dy/dt)^2) dt

The Attempt at a Solution

when n=2, it is just a regular circle, therefore it is not too difficult to parametrize the function (for example, x=cost y=cost)
However, when n gets other even number integers, i do not know how i can define it in parametrics.

 

[Redbelly98]

Homework Statement

The curves with equations x" + y" = 1, n = 4, 6, 8, , are called fat circles.

I take it that should be

xⁿ + yⁿ = 1​

Graph the curves with n = 2, 4, 6, 8, and 10 to see why. Set up an integral for the length S(2k), the arc of the fat circle with n=2k. Without attempting to evaluate the integral, state the value of lim S(2k) as x approaches infinity.

Homework Equations

Formula for the lengths of curves.
L= intergal of square root ((dx/dt)^2)/(dy/dt)^2) dt

That's not the correct formula for arc length of a curve. Moreover, it is not necessary to paramaterize x and y in terms of t. You can read more about the arc length formula at wikipedia:

http://en.wikipedia.org/wiki/Arc_length

 

[hangainlover]

sorry i apologize...too many typos ...
yeah that should be n and
the length formula should be square root of ((dy/dt)^2 + (dx/dt)^2 )dt

 

[Redbelly98]

Okay, makes more sense now.

The wiki article has the arc length in terms of dy/dx. I would try that form instead.

Disclaimer: I haven't carried this through to actually solving the problem.

 

[hangainlover]

I've been actually trying that way.
I found that implicit differentiation doesn't help us solve it.
But, we can still isolate y by taking root of the rest.

However, I need to come up with some generalization to evaluate the value of lim S(2k) as x approaches infinity.

 

[Redbelly98]

The problem explicitly says, do not attempt to evaluate the integral.

Try graphing the curve for several values of n. What shape does the curve approach, for large values of n? You need to make the graph and have a look at it.