Answer
The length: $s = 5\left( {{\rm{e}} - 1} \right)$
Work Step by Step
We have
$x\left( t \right) = 3{{\rm{e}}^t} - 3$, ${\ \ }$ $x'\left( t \right) = 3{{\rm{e}}^t}$,
$y\left( t \right) = 4{{\rm{e}}^t} + 7$, ${\ \ }$ $y'\left( t \right) = 4{{\rm{e}}^t}$.
By Theorem 1 of Section 12.2, the length of the curve for $0 \le t \le 1$ is
$s = \mathop \smallint \limits_0^1 \sqrt {{{\left( {3{{\rm{e}}^t}} \right)}^2} + {{\left( {4{{\rm{e}}^t}} \right)}^2}} {\rm{d}}t = \mathop \smallint \limits_0^1 \sqrt {9{{\rm{e}}^{2t}} + 16{{\rm{e}}^{2t}}} {\rm{d}}t$
$s = 5\mathop \smallint \limits_0^1 {{\rm{e}}^t}{\rm{d}}t = 5{{\rm{e}}^t}|_0^1 = 5\left( {{\rm{e}} - 1} \right)$
