Question

Find the length of the curve.

Short answer

The required length of the given curve is\(\left( {\frac{\pi }{2} - \frac{{3\sqrt 3 }}{8}} \right)\).

Step-by-step solution

Definition of curve

A curve is a mathematical entity that is similar to a line but does not have to be straight.

Find the limit L

Consider the given equations.

We know that the length \(L\)of a curve,

\(L = \int_a^b {\sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2}} } dt\)

Hence, L\( = {\sin ^4}\left( {\frac{\theta }{3}} \right)\)

Change the limit into integration

Let us solve the given problem.

Length is:

\(\begin{aligned}{c}L = \int_0^\pi {\sqrt {{{\sin }^4}\left( {\frac{\theta }{3}} \right)} } d\theta \\ = \int_0^\pi {{{\sin }^2}} \left( {\frac{\theta }{3}} \right)d\theta \\ = \int_0^\pi {\frac{{1 - \cos \left( {\frac{{2\theta }}{3}} \right)}}{2}} d\theta \\ = \frac{1}{2}\int_0^\pi d \theta - \frac{1}{2}\int_0^\pi {\cos } \left( {\frac{{2\theta }}{3}} \right)d\theta \\ = \left. {\frac{1}{2}\theta } \right|_0^\pi - \left. {\frac{1}{2} \cdot \frac{3}{2}\sin \left( {\frac{{2\theta }}{3}} \right)} \right|_0^\pi \\ = \frac{\pi }{2} - \frac{{3\sqrt 3 }}{8}\end{aligned}\)

Therefore, the required length of the curve is\(\left( {\frac{\pi }{2} - \frac{{3\sqrt 3 }}{8}} \right)\).