Question

Find the length of the curve.

Short answer

Length of the given curve is\(\left( {\sinh 3} \right)\).

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 \({\bf{L}}\)

Consider the given equations is

We know that the length \(L\)of a curve is given by

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

Now,

\(\begin{aligned}{c}\frac{{dx}}{{dt}} = 3\\\frac{{dy}}{{dt}} = 3\sinh t\end{aligned}\)

\(\begin{aligned}{c}L = \int_0^1 {\sqrt {{{(3)}^2} + {{\left( {3{{\sinh }^2}3t} \right)}^2}} } dt\\ = \int_0^1 {\sqrt {9 + 9{{\sinh }^2}3t} } dt\\ = \int_0^1 {\sqrt {9\left( {1 + {{\sinh }^2}3t} \right)} } dt\\ = \int_0^1 {\sqrt {9{{\cosh }^2}3t} } dt\\ = \int_0^1 3 \cosh tdt\end{aligned}\)

Change the limit of integration:

\((\sinh 3t)_0^1 = \sinh 3\)

Hence, the required length of the curve is \(L = \sinh 3\).