Find the exact area of the surface obtained by rotating the given curve about the x-axis. $x = 2 \cos ^ { 3 } \theta , \quad y = 2 \sin ^ { 3 } \theta , \quad 0 \leq \theta \leq \pi / 2$
A) $\frac { 24 \pi } { 5 }$
B) $\frac { 18 \pi } { 5 }$
C) $\frac { 12 \pi } { 5 }$
D) $\frac { 2 \pi } { 4 }$
E) None of these

Question

Quizplus · Accepted Answer

The curve is a parametric form of an astroid. The surface area of revolution about the x-axis is given by $ \int 2\pi y \sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2} \, d\theta $. Calculating this integral gives $\frac{24\pi}{5}$.

Find the Exact Area of the Surface Obtained by Rotating x=2cos⁡3θ,y=2sin⁡3θ,0≤θ≤π/2x = 2 \cos ^ { 3 } \theta , \quad y = 2 \sin ^ { 3 } \theta , \quad 0 \leq \theta \leq \pi / 2x=2cos3θ,y=2sin3θ,0≤θ≤π/2

Find the Exact Area of the Surface Obtained by Rotating $x = 2 \cos ^ { 3 } \theta , \quad y = 2 \sin ^ { 3 } \theta , \quad 0 \leq \theta \leq \pi / 2$