Chapter 13 - Vector Functions - 13.3 Arc Length and Curvature - 13.3 Exercises - Page 908: 6

$42$

As we are given that $r(t)=t^2i+9tj+4t^{3/2}k$ ; $1 \leq t \leq 4$ Length of the curve can be obtained by using formula, such as $L=\int_a^b |r'(t)| dt$ Now, $r'(t)=\lt 2t,9,6t^{1/2}\gt$ ; $|r'(t)|=\sqrt {( 2t)^2+(9)^2+(6t^{1/2})^2}dt$ $=\sqrt {4t^2+81+36t}dt$ $=2t+9$ From formula: $L=\int_{1}^4(2t+9) dt=(t^2+9t)|_{1}^4=((4)^2-(1)^2)+(9(4)-9(1))=42$