- #1
PFuser1232
- 479
- 20
I have a few questions about finding volumes of solids of revolution (in a typical first year single variable calculus course).
1) I can rotate any region about any horizontal/vertical axis. How exactly do I rotate a region about a line that is neither horizontal nor vertical (##y = x - 1## for example)? Or is this beyond Calculus I/II?
2) Every solids of revolution example I have come across so far always involved an axis of rotation that does not pass through the region (which is to be rotated). How can one find the volume of the solid obtained when a region is rotated about an axis passing through the region itself?
3) The method of "disks/washers" fails in cases where, say, ##y## is given as a function of ##x## and it's impossible to find ##x## as a function of ##y## and we're supposed to rotate a region about a vertical axis, in which case we resort to the "shell" method. Does the shell method ever fail? Or does it always work?
1) I can rotate any region about any horizontal/vertical axis. How exactly do I rotate a region about a line that is neither horizontal nor vertical (##y = x - 1## for example)? Or is this beyond Calculus I/II?
2) Every solids of revolution example I have come across so far always involved an axis of rotation that does not pass through the region (which is to be rotated). How can one find the volume of the solid obtained when a region is rotated about an axis passing through the region itself?
3) The method of "disks/washers" fails in cases where, say, ##y## is given as a function of ##x## and it's impossible to find ##x## as a function of ##y## and we're supposed to rotate a region about a vertical axis, in which case we resort to the "shell" method. Does the shell method ever fail? Or does it always work?