Paul's Online Notes
Home / Calculus II / Parametric Equations and Polar Coordinates / Arc Length with Parametric Equations
Show Mobile Notice Show All Notes Hide All Notes
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 views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 3-4 : Arc Length with Parametric Equations

For all the problems in this section you should only use the given parametric equations to determine the answer.

For problems 1 and 2 determine the length of the parametric curve given by the set of parametric equations. For these problems you may assume that the curve traces out exactly once for the given range of t’s.

1. $$x = 8{t^{\frac{3}{2}}}\hspace{0.25in} y = 3 + {\left( {8 - t} \right)^{\frac{3}{2}}}\hspace{0.25in} 0 \le t \le 4$$ Solution
2. $$x = 3t + 1\hspace{0.25in} y = 4 - {t^2}\hspace{0.25in} - 2 \le t \le 0$$ Solution
3. A particle travels along a path defined by the following set of parametric equations. Determine the total distance the particle travels and compare this to the length of the parametric curve itself. $x = 4\sin \left( {\frac{1}{4}t} \right)\hspace{0.25in}y = 1 - 2{\cos ^2}\left( {\frac{1}{4}t} \right)\hspace{0.5in} - 52\pi \le t \le 34\pi$ Solution

For problems 4 and 5 set up, but do not evaluate, an integral that gives the length of the parametric curve given by the set of parametric equations. For these problems you may assume that the curve traces out exactly once for the given range of t’s.

1. $$x = 2 + {t^2}\hspace{0.25in}y = {{\bf{e}}^t}\sin \left( {2t} \right)\hspace{0.25in}\,0 \le t \le 3$$ Solution
2. $$\displaystyle x = {\cos ^3}\left( {2t} \right)\hspace{0.25in} y = \sin \left( {1 - {t^2}} \right)\hspace{0.25in}\, - \frac{3}{2} \le t \le 0$$ Solution