Answer
$ 18.6833$
Work Step by Step
Given: $r(t)=\lt t^2,t^3,t^4 \gt$; $0 \leq t \leq 2$
To calculate the length of the curve we will have to use the formula:
$L=\int_a^b |r'(t)| dt$
Thus,
$r'(t)=\lt 2t,3t^2,4t^3\gt$
and $|r'(t)|=\sqrt {( 2t)^2+(3t^2)^2+(4t^3)^2}dt$
$=\sqrt{ 4t^2+9t^4+16t^6}$
$L=\int_{0}^2(\sqrt{ 4t^2+9t^4+16t^6}) dt$
As per question, we will use calculator to find the length of the curve.
Hence, $L= 18.6833$
