Question

Graph the curve with parametric equations \(x = cost,\;y = sin3t,\;z = sint\). Find the total length of this curve correct to four decimal places.

\(L \approx 13.9744\)s

Short answer

The length of the given curve is, .

Step-by-step solution

Step 1: Given parametric equations

The parametric equations for the given curve are given as

\(\begin{aligned}{l}x = \cos t\\y = \sin 3t\\z = \sin t\end{aligned}\)

Using above mentioned parametric equations we get

\(\begin{aligned}{l}\frac{{dx}}{{dt}} = - \sin t\\\frac{{dy}}{{dt}} = 3\cos 3t\\\frac{{dz}}{{dt}} = \cos t\end{aligned}\)

Step 2: The graph

The graph for the given curve can be represented as:

From the graph plot we see that the curve has range from \(0\;to\;2\pi \). Therefore, the length of the curve from \(t = 0\;to\;t = 2\pi \)is estimated as

\(\begin{aligned}{l}L = \int\limits_0^{2\pi } {\sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2} + {{\left( {\frac{{dz}}{{dt}}} \right)}^2}} \;} dt\\L = \int\limits_0^{2\pi } {\sqrt {{{\sin }^2}t + 9{{\cos }^2}3t + {{\cos }^2}t} \;dt} \\L = \int\limits_0^{2\pi } {\sqrt {1 + 9{{\cos }^2}3t} } \;dt\end{aligned}\)

Step 3: Use online tool for calculating the integral

Calculate using Online CAS we have

\(\begin{aligned}{l}L = \int\limits_0^{2\pi } {\sqrt {1 + 9{{\cos }^2}3t} } \;dt\\L \approx 13.974417\\L \approx 13.9744\end{aligned}\)