Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 9B (Vector Calculus)

For a regular parametrized curve $r(t)$, the Frenet frame isEquivalently, wherever $dT/dt\ne 0$, one may write $N=(dT/dt)/|dT/dt|$ and $B=T\times N$.

Differentiation givesand $|r^{\prime }|=6|\sin t\cos t|$. ThusThe plus sign applies on $(0,\pi /2)\cup (\pi ,3\pi /2)$ and the minus sign on $(\pi /2,\pi )\cup (3\pi /2,2\pi )$.

On each open quadrant,The constant binormal proves that the curve is planar; indeed $z=\sqrt{3}y$.

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