Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 9B / c / Solution

For $n=(a,b,c)$ and $r=(x,y,z)$, the integrand is $\frac{1}{2}(n\times r)\cdot dr$. Sincethe Stokes theorem gives the signed planar area, and its absolute value gives the ordinary area.

The $xy$ projection of the given curve is the astroid $x=2{\cos }^{3}t$, $y={\sin }^{3}t$, whose area isThe plane $z=\sqrt{3}y$ stretches area from its $xy$ projection by $\sqrt{1+3}=2$. Hence