Find out the surface area of the solid acquired by rotating the following parametric curve about the x-axis.

x = cos³ θ

y = sin³ θ

 0 ≤ θ ≤ ?/2

Solution

We will first require the derivatives of the parametric equations.

dx/dt =  -3 cos² θ sin θ

 dy/dt = 3sin² θ cos θ

 Previous to plugging into the surface area formula let us get the ds out of the way.

ds = √ (9 cos4 θsin2 θ + 9 sin4 θ cos2 θ) dt

= 3|cosθ sin θ| √ (cos² θ + sin² θ)

=3 cos θ sin θ

Note that we could drop the absolute value bars as both sine and cosine are positive in this range of θ given.

 Now let us get the surface area and do not forget to as well plug in for the y.