Question

Use a calculator to find the length of the curve correct to four decimal places. If necessary, graph the curve to deter-mine the parameter interval \({\rm{r = sin(\theta /4)}}\).

Short answer

\( \approx {\rm{17}}{\rm{.1568}}\)

Step-by-step solution

Graph

Evaluation.

\(\begin{aligned}{c}{\rm{ r = sin}}\frac{{\rm{\theta }}}{\begin{aligned}{l}{\rm{4}}\\\end{aligned}}\\{\rm{Keep in mind that the duration of sin (bx) is}}\\\frac{{{\rm{2\pi }}}}{{\rm{b}}}{\rm{ = }}\frac{{{\rm{2\pi }}}}{{\frac{{\rm{1}}}{{\rm{4}}}}}{\rm{ = 8\pi }}\\{\rm{As a result, the limitations will be 0 to 8\pi }}\\{\rm{The arc length will necessitate }}r\prime \\r\prime {\rm{ = }}\frac{{\rm{1}}}{{\rm{4}}}{\rm{cos}}\frac{{\rm{\theta }}}{{\rm{4}}}\\\\{\rm{Use the magic calculator to plug into the arc length formula}}{\rm{.}}\\{\rm{L = }}\int_{\rm{a}}^{\rm{b}} {\sqrt {{{\rm{r}}^{\rm{2}}}{\rm{ + }}{{\left( {r\prime } \right)}^{\rm{2}}}} } {\rm{d\theta }}\\{\rm{ = }}\int_{\rm{0}}^{{\rm{8\pi }}} {\sqrt {{\rm{si}}{{\rm{n}}^{\rm{2}}}\frac{{\rm{\theta }}}{{\rm{4}}}{\rm{ + }}{{\left( {\frac{{\rm{1}}}{{\rm{4}}}{\rm{cos}}\frac{{\rm{\theta }}}{{\rm{4}}}} \right)}^{\rm{2}}}} } {\rm{d\theta }}\\\\\end{aligned}\)

\( \approx {\rm{17}}{\rm{.1568}}\).