find the length of$\displaystyle x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$
?
i use the parametrisation
y=a(sint)^3
x=a(cost)^3
but what is the range of t?
In the first quadrant the curve looks like this: plot x^(2/3) + y^(2/3) = 1 - Wolfram|Alpha
I suggest you find the arclength for the curve in the first quadrant - the domain for t should be obvious - and then multiply by 4.
You've changed the question so many times, it is hard to be sure what you are asking.
Surely you know that if you are writing x and y in terms of parameter t, then the arclength is given by
$\displaystyle \int \sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dt}\right)^2} dt$
That is an integral and there are no partial derivatives.
You said you wanted to use the parameterization y=a(sint)^3, x=a(cost)^3.
In the graph Mr. Fantastic showed you initially, x goes from 0 up to 1 while y goes from 1 down to 0. (He set a= 1.)
For what t is $\displaystyle x= a(cos t)^3= 0$? For what t is $\displaystyle a(cos t)^3= a$?