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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.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 9.5 : 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.
- Rotate \(x = {t^2} - 3\hspace{0.25in}y = 2 + {t^2}\hspace{0.25in}0 \le t \le 5\) about the \(x\)-axis.
- Rotate \(x = - 8t\hspace{0.25in}y = 6 + {t^2}\hspace{0.25in} - 3 \le t \le 0\) about the \(y\)-axis.
- Rotate \(x = {t^2}\hspace{0.25in}y = {t^4} - 2\hspace{0.25in}0 \le t \le 2\) about the \(y\)-axis.
- 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.
- 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.
- 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.
- 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.