Question

Use a calculator to find the length of the curve correct to four decimal places. If necessary, graph the curve to determine the parameter interval.

\({\rm{r = sin(6sin\theta )}}\)

Short answer

The length of the curve is \({\rm{\gg 8}}{\rm{.0091}}\)

Step-by-step solution

Graph is representing the given equation.

Solve the equation.

Pink graph,

\({\rm{r = sin(6sin\theta )}}\)

Complete figure over the interval\({\rm{(0,}}\pi {\rm{)}}\)

\(\frac{{{\rm{dr}}}}{{{\rm{d\theta }}}}{\rm{ = cos(6sin\theta ) \times 6cos\theta }}\)

Polar arc length formula.

Use the polar arc length formula,

\(\begin{aligned}{c}{\rm{L = }}\int_{\rm{a}}^{\rm{b}} {\sqrt {{{\rm{r}}^{\rm{2}}}{\rm{ + }}{{\left( {\frac{{{\rm{dr}}}}{{{\rm{d\theta }}}}} \right)}^{\rm{2}}}} } {\rm{d\theta }}\\{\rm{ = }}\int_{\rm{0}}^\pi {\sqrt {{{{\rm{(sin(6sin\theta ))}}}^{\rm{2}}}{\rm{ + (6cos\theta cos(6sin\theta )}}{{\rm{)}}^{\rm{2}}}} } {\rm{d\theta }}\\{\rm{ = }}\int_{\rm{0}}^\pi {\sqrt {{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{(6sin\theta ) + 36co}}{{\rm{s}}^{\rm{2}}}{\rm{\theta co}}{{\rm{s}}^{\rm{2}}}{\rm{(6sin\theta )}}} } {\rm{d\theta }}\\{\rm{\gg 8}}{\rm{.0091}}\end{aligned}\)

The length of the curve is \({\rm{\gg 8}}{\rm{.0091}}\)