Answer
$42$
Work Step by Step
Given: $r(t)=t^2i+9tj+4t^{3/2}k$ ; $1 \leq t \leq 4$
To calculate the length of the curve we will have to use the formula:
$L=\int_a^b |r'(t)| dt$
Thus,
$r'(t)=\lt 2t,9,6t^{1/2}\gt$
and $|r'(t)|=\sqrt {( 2t)^2+(9)^2+(6t^{1/2})^2}dt$
$=\sqrt {4t^2+81+36t}dt$
$=2t+9$
$L=\int_{1}^4(2t+9) dt$
$\implies L=(t^2+9t)|_{1}^4$
$=((4)^2-(1)^2)+(9(4)-9(1))$
$=42$
Hence, $L=42$