Concept:
The length of the curve in three-dimensional space given in the parametric form is
\(L = \mathop \smallint \limits_a^b \sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2} + \;{{\left( {\frac{{dz}}{{dt}}} \right)}^2}} \;.dt,\;a \le t \le b\)
And the length of the curve in two- dimensional space given in parametric form is
\(L = \mathop \smallint \limits_a^b \sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2}\;} \;.dt,\;a \le t \le b\)
Calculation:
Given, \(x\left( t \right) = cost,\;y\left( t \right) = \;sint,\;z\left( t \right) = \frac{2}{\pi }t,\;0 \le t \le \frac{\pi }{2}\;\;\)
\(L = \mathop \smallint \limits_0^{\pi /2} \sqrt {{{\left| {\frac{{d\left( {cost} \right)}}{{dt}}} \right|}^2} + {{\left| {\frac{{d\left( {sint} \right)}}{{dt}}} \right|}^2} + {{\left| {\frac{2}{\pi }\frac{{d\left( t \right)}}{{dt}}} \right|}^2}\;} .dt\;\)
\( = \mathop \smallint \limits_0^{\pi /2} \sqrt {{{\sin }^2}t + co{s^2}t + \frac{4}{{{\pi ^2}}}} \;.dt\)
\( = \mathop \smallint \limits_0^{\pi /2} \sqrt {1 + \frac{4}{{{\pi ^2}}}} .dt\)
\( = 1.185\mathop \smallint \limits_0^{\pi /2} dt = 1.185 \times \frac{\pi }{2}\)
= 1.862
≈ 1.86
