Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.
Section 12.9 : Arc Length with Vector Functions
1. Determine the length of \(\vec r\left( t \right) = \left( {3 - 4t} \right)\vec i + 6t\,\vec j - \left( {9 + 2t} \right)\vec k\) from \( - 6 \le t \le 8\).
Show All Steps Hide All Steps
Start SolutionWe first need the magnitude of the derivative of the vector function. This is,
\[\vec r'\left( t \right) = - 4\vec i + 6\,\vec j - 2\vec k\] \[\left\| {\vec r'\left( t \right)} \right\| = \sqrt {16 + 36 + 4} = \sqrt {56} = 2\sqrt {14} \] Show Step 2The length of the curve is then,
\[L = \int_{{ - 6}}^{8}{{2\sqrt {14} \,dt}} = \left. {2\sqrt {14} t} \right|_{ - 6}^8 = \require{bbox} \bbox[2pt,border:1px solid black]{{28\sqrt {14} }}\]