Question

Tough Calc Question:
Let r>0. The equations below parametrize an astroid

x= r(cos^3)t and y= r(sin^3)t for 0<=t<=2pi

a. find the length of the asteroid
b. show that the astroid is alternatively the graph of
x^(2/3) + y^(2/3) = r^(2/3)

c. Find the area S of the surface generated by revolving C about the x axis.

Been stuck on this problem for a while.. any help would be appreciated!

Thanks.

Answer 1

(A)
\[\text{length of curve}=\Large \int_0^{2\pi}\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\;\mathrm dt\]

Answer 2

(B)
\[\Large {x=r\cos^3t\\x^{1/3}=r^{1/3}\cos t\\x^{2/3}=r^{2/3}\cos^2 t\\}\]
do the same on \(y\), and use the trigonometric identity \[\Large \cos^2 t + \sin^2 t=1\]

Answer 3

C
\[S = \int\limits_{0}^{2 \pi} 2 \pi y \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\:dt\]