Question

Find the length of the parametric curve
x=cos^3t,y=sin^3t,0<t<pi/2

Answer 1

Answer on Question#50337 - <math> - <other>

Find the length of the parametric curve $\left\{ \begin{array}{ll}x(t) = \cos^3 t\\ y(t) = \sin^3 t \end{array} \right.$ $0 < t < \frac{\pi}{2}$

Solution. Since, the length of the parametric curve is $L = \int_{a}^{b}\sqrt{(x'(t))^2 + (y'(t))^2} dt$ then:

$$\begin{array}{l} L = \int_ {0} ^ {\pi / 2} \sqrt {(x ^ {\prime} (t)) ^ {2} + (y ^ {\prime} (t)) ^ {2}} d t = \int_ {0} ^ {\pi / 2} \sqrt {(- 3 \cos^ {2} t \sin t) ^ {2} + (3 \sin^ {2} t \cos t) ^ {2}} d t = \int_ {0} ^ {\pi / 2} \sqrt {9 \cos^ {4} t \sin^ {2} t + 9 \sin^ {4} t \cos^ {2} t} d t = \\ = 3 \int_ {0} ^ {\pi / 2} \sqrt {\cos^ {2} t \sin^ {2} t (\cos^ {2} t + \sin^ {2} t)} d t = 3 \int_ {0} ^ {\pi / 2} \sqrt {\cos^ {2} t \sin^ {2} t} d t = 3 \int_ {0} ^ {\pi / 2} | \cos t \sin t | d t = 3 \int_ {0} ^ {\pi / 2} \cos t \sin t d t = \\ = 3 \int_ {0} ^ {\pi / 2} \sin t d (\sin t) = 3 \frac {\sin^ {2} t}{2} \Bigg | _ {0} ^ {\pi / 2} = \frac {3}{2} \\ \end{array}$$

Answer: The length of the parametric curve $\left\{ \begin{array}{ll}x(t) = \cos^3 t\\ y(t) = \sin^3 t \end{array} \right.$ $0 < t < \frac{\pi}{2}$ is $\frac{3}{2}$

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