Only to be done in "Mathematica" and no other language. Please do exercise $1$ & $2.$ Check the picture for clarificaiton. I'll thumbs up your effort.

Modules for Exercise $2$

required. The output is a graph of the space curve $y$ that is represented by the parametric functions along with the superimposed inscribed polygon that consists of $n$ line segments.

the curve.

Exercise $2.$

length of the curve using the arc length formula and compare with the approximation.

The astroid given by the parametric equations $x(t)=si{n}^{3}t,y(t)=co{s}^{3}t,z(t)=cos(2t)$ in the interval $0,2$ with $n=8,16.$

The curve given by the parametric equations $x(t)=cost,y(t)=cos(3t),z(t)=sint$ in the interval $0,2$ with $n=8,32.$

The curve given by the parametric equations $x(t)=cost,y(t)=cos(3t),z(t)=sint$ in the interval $0,4$ with $n=16,32.$

Compare the results of parts $2)$ and $3).$ Explain why the lengths of seemingly the same curves differ so much.