Student handout: Curvilinear Volume Elements
Vector Calculus I 2022
Students practice infinitesimal reasoning in cylindrical and spherical coordinates.
- group Small Group Activity 30 min. whiteboards/markers/erasers, handout, coordinate axes in the ceiling, pumpkins, canned pineapple slices Student handout (PDF)
What students learn
Students practice infinitesimal reasoning in cylindrical and spherical coordinates.
Cylindrical Coordinates
Using the first figure below, determine the length \(d\ell\) of each of the three paths shown (the three thick lines). Notice that, along any of these three paths, only one coordinate \(s\), \(\phi\), or \(z\) is changing at a time (i.e. along path 1, \(dz\ne0\), but \(d\phi=0\) and \(ds=0\)).Path 1: \(d\ell=\)
Path 2: \(d\ell=\)
Path 3: \(d\ell=\)
Use your results to determine the volume of the region.
\(d\tau=\)
Spherical Coordinates
Using the second figure below, determine the length \(d\ell\) of each of the three paths shown (the three thick lines). Notice that, along any of these three paths, only one coordinate \(r\), \(\theta\), or \(\phi\) is changing at a time (i.e. along path 1, \(d\theta\ne0\), but \(dr=0\) and \(d\phi=0\)).
(Be careful: One path is trickier than the others.)
Path 1: \(d\ell=\)
Path 2: \(d\ell=\)
Path 3: \(d\ell=\)
Use your results to determine the volume of the region.
\(d\tau=\)
- Author Information
- Corinne Manogue, Tevian Dray, & Katherine Meyer
- Keywords
- Learning Outcomes
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