Question

Find the exact length of the polar curve.

\({\rm{r = 3sin\theta ,0}} \le {\rm{\theta }} \le {\rm{\pi /3}}\)

Short answer

The length of the polar curve is\({\rm{\pi }}\).

Step-by-step solution

Defining Length.

The length of the polar curve is given by

\({\rm{L = }}\int_{\rm{a}}^{\rm{b}} {\sqrt {{{\rm{r}}^{\rm{2}}}{\rm{ + }}{{\left( {\frac{{{\rm{dr}}}}{{{\rm{\;d\theta }}}}} \right)}^{\rm{2}}}} } \)

Integrating the length.

\(\begin{aligned}{c}{\rm{L = }}\int_{\rm{0}}^{\frac{{\rm{\pi }}}{{\rm{3}}}} {\sqrt {{{{\rm{(3sinq)}}}^{\rm{2}}}{\rm{ + (3cosq}}{{\rm{)}}^{\rm{2}}}} } {\rm{dq}}\\{\rm{ = }}\int_{\rm{0}}^{\frac{{\rm{\pi }}}{{\rm{3}}}} {\sqrt {{\rm{9}}\left( {{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{q + si}}{{\rm{n}}^{\rm{2}}}{\rm{q}}} \right)} } {\rm{dq}}\\{\rm{ = }}\int_{\rm{0}}^{\frac{{\rm{\pi }}}{{\rm{3}}}} {\rm{3}} {\rm{dq}}\\{\rm{ = }}\left. {{\rm{3q}}} \right|_{\rm{0}}^{\frac{{\rm{\pi }}}{{\rm{3}}}}\\{\rm{ = \pi }}\end{aligned}\)

\({\rm{L = \pi }}\)

Therefore, the length of the curve is\({\rm{\pi }}\).