Surface Area of an Astroid

Hi all,

I have a presentation tomorrow on a homework problem and I am stuck on the following problem:

Find the surface area of the astroid x^(2/3) + y^(2/3) = 1

Since it is symmetric I leave the bottom part out and revolve the upper part around the y-axis and multiply the result by 2. Limits of integration is from 0 to 1.

So far I think I am on the right track, but when I try to integrate this it gets really ugly.

I transformed the above equation into x=f(y) notation (since it is rotated around y-axis) and got y= (1-x^(2/3))^(3/2)

When plugged into the calculator I get a somewhat reasonable answer of 2Pi for the surface area of the whole asteroid.

Does anyone have any suggestions on the integration (I tried several different things but always get stuck)?

Thanks