Volumes of Revolution - A Methodical Approach
This notebook will demonstrate many permutations of rotating the above regions about various axes using disk, washer and shell methods. Depending on the goal, we will integrate either with respect to x- or with respect to y.
Rotate regions I, II, and III about the x axis.
Region 1 is bounded by the curve above, the x axis below, and the line . Here is the surface of the volume created.
Using the disk method, with the disk perpendicular to the x-axis,
Region 2 is bounded by the curve below, the y axis, and the line above. Here is the surface of the volume created.
Rotating region II about the x axis requires use of the washer method, i.e. one disk subtracted from another.
Region 3 is bounded above by the curve and below by the curve . Here the surface of the volume created.
Using the washer method, with the washer / disk perpendicular to the x-axis,
Rotate regions I, II, and III about the y axis.
Region I, bounded by , the x axis, and the line , rotated about the y axis, looks like this:
Using the shell method, the area of each shell is
Region II, bounded by , the y axis, and the line , rotated about the y axis.
Using the shell method again,
Region III, rotated about y.
The next group will be the same regions and rotations, but replacing the shell method with the disk or washer and integrating with respect to y. When integrating with respect to y we'll need to use the inverse functions.
Rotating region I around y, we get a washer with
Rotating region II about y and integrating with respect to y yields a simple disk with
And, rotating region III about y and integrating with respect to y gives a washer with
Rotate regions I, II, and III about y=1
Rotating region I about y=1 using the washer method, we have
Rotating region II about y=1 is a simple disk method with .
For region III about y=1 we have a washer with
Rotate regions I, II, and III about x=1
Rotating region I about x=1 and using the shell method yields
Rotating region II about x=1, also using shells.
Region III rotated about x=1.
Rotating region III about x=1 using shells.