Area and Arc Length of |y|^n+|x|^n=a^n

**john f** · December 3rd 2009, 01:49 PM

I've been trying to fing both the area and the arc length of
$|y|^n+|x|^n=a^n$
with respect to x where a and n are constants

Graphing examples of the function as well as solving simple examples where a=1 and n=1 (diamond) or n=2 (circle) has shown that the graph is symetrical in all quadrants.

I've written the in equation terms of y and set limits to account for the symetry and have gotten.
$A(x)=4 \int_0^a (a^n-x^n)^{(1/n)}$
for the area and
$L(x)=4 \int_0^a \sqrt{1+(d/dx(a^n-x^n)^{(1/n)})}$
But I do not how to procede from there.

**john f** · December 4th 2009, 11:51 AM

I think the power of the integral becomes (n+1)/n but i don't know how to handle the (a^n-x^n) portion

**john f** · December 6th 2009, 04:50 PM

Anyone? I still can't figure it out.

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