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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.
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