Question

Find the volume of the solid generated by revolution of x = acos3$\theta$, y = sin3$\theta$ about its x- axis.

Answer 1

The standard formula for rotation about the x-axis, through an angle of 2π-radians, when x and y are given as functions of θ is

A=2π ∫ θ2θ1y(dxdθ)2+(dydθ)2−−−−−−−−−−−−−√ dθ

In your case x=acos3θ and y=asin3θ. It follows that

(dxdθ)2+(dydθ)2==≡(−3acos2θsinθ)2+(3asin2θcosθ)9a2cos4θsin2θ+9a2sin4θcos2θ9a2cos2θsin2θ

Since 0≤θ≤π we need to take care because cosθ changes sign. We have

A==2π∫π0asin3θ9a2cos2θsin2θ−−−−−−−−−−−−√ dθ6πa2∫π0sin3θ⋅|cosθ|⋅|sinθ| dθ

Since sinθ≥0 for all 0≤θ≤π we have |sinθ|≡sinθ. However, cosθ≥0 for all 0≤θ≤π2 and cosθ≤0 for all π2≤θ≤π. Hence |cosθ|≡cosθ for all 0≤θ≤π2, while |cosθ|≡−cosθ for all π2≤θ≤π. It follows that

A===6πa2∫π/20sin4θcosθ dθ−6πa2∫ππ/2sin4θcosθ dθ6πa2[15sin5θ]π/20−6πa2[15sin5θ]ππ/2125