Paul's Online Notes
Paul's Online Notes
Home / Calculus II / Parametric Equations and Polar Coordinates / Surface Area with Parametric Equations
Show General Notice Show Mobile Notice Show All Notes Hide All Notes
General Notice

This is a little bit in advance, but I wanted to let everyone know that my servers will be undergoing some maintenance on May 17 and May 18 during 8:00 AM CST until 2:00 PM CST. Hopefully the only inconvenience will be the occasional “lost/broken” connection that should be fixed by simply reloading the page. Outside of that the maintenance should (fingers crossed) be pretty much “invisible” to everyone.

Paul
May 6, 2021

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.
Assignment Problems Notice
Please do not email me to get solutions and/or answers to these problems. I will not give them out under any circumstances nor will I respond to any requests to do so. The intent of these problems is for instructors to use them for assignments and having solutions/answers easily available defeats that purpose.

Section 3-5 : Surface Area with Parametric Equations

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

For problems1 – 4 determine the surface area of the object obtained by rotating the parametric curve about the given axis. For these problems you may assume that the curve traces out exactly once for the given range of \(t\)’s.

  1. Rotate \(x = {t^2} - 3\hspace{0.25in}y = 2 + {t^2}\hspace{0.25in}0 \le t \le 5\) about the \(x\)-axis.
  2. Rotate \(x = - 8t\hspace{0.25in}y = 6 + {t^2}\hspace{0.25in} - 3 \le t \le 0\) about the \(y\)-axis.
  3. Rotate \(x = {t^2}\hspace{0.25in}y = {t^4} - 2\hspace{0.25in}0 \le t \le 2\) about the \(y\)-axis.
  4. Rotate \(x = 2 + t\hspace{0.25in}y = 4{{\bf{e}}^{ - \,\,\frac{1}{2}t}}\hspace{0.25in} - 1 \le t \le 2\) about the \(x\)-axis.

For problems 5 – 7 set up, but do not evaluate, an integral that gives the surface area of the object obtained by rotating the parametric curve about the given axis. For these problems you may assume that the curve traces out exactly once for the given range of \(t\)’s.

  1. Rotate \(x = 2 + {{\bf{e}}^{\cos \left( t \right)}}\hspace{0.25in}y = 1 + {t^2}\hspace{0.25in} - 2 \le t \le 0\) about the \(x\)-axis.
  2. Rotate \(x = {\cos ^2}\left( t \right)\hspace{0.25in}y = 2\cos \left( {2t} \right) - \sin \left( t \right)\hspace{0.25in}0 \le t \le 1\) about the \(y\)-axis.
  3. Rotate \(x = {t^2}\hspace{0.25in}y = \ln \left( {3 + {{\bf{e}}^{ - t}}} \right)\hspace{0.25in}0 \le t \le 2\) about the \(x\)-axis.