Question

Find the arc length parametrization of the curve r(t) = < cost, sint, 2/3t3/2 >, with the parameter s measuring from (1, 0, 0).

Answer 1

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Answer 2

Given :

The curve with an equation r(t)= <cost, sint, $\frac{2}{3}t$>

To find :

The arc length has to be found.

Step-by-step explanation:

• The arc length can be found by following these steps.
• It is given that the parameters are $(1,0,0)$.
• Firstly, the differentiation of $r(t)$ must be found.
• The differentiation would be

        $r'(t)=$

• Then, take the square of the equation.
• Square of the equation will be,

        $|r'(t)|=\sqrt{sin^{2} t+cos^{2} t+1}$

• The formula used is

        $sin^{2}t+ cos^{2}t=1$

• Substitute the formula in the above equation.
• Then, the equation will become

        $|r'(t)|=\sqrt{1+1}$

• Sum it all up

        $|r'(t)|=\sqrt{2}$

• Now take the integral of the equation,

        $r(t)=$∫$\sqrt{2}$

• Take the integration with the limits from $0$ to $2\pi$.
• Because, the angles will exist from $0$ to $2\pi$. After $2\pi$ the values will repeat.

• After taking the integral, the answer would be

        $2\sqrt{2} \pi$      

Final answer :

The length of the arc is $2\sqrt{2} \pi$ $units$.