13. Volume
Recall: The Volume of a Solid computed using Thin Disks is
c1b. Volume using Thin Disks - Example 2
Now we come to volumes of revolution which do not have a classical formula:
The region between the curve and the -axis for is rotated about the -axis. Find the volume swept out.
The curve is a function of ; so this is a -integral and we cut up the -axis. Here is the region and one rotated disk:
The radius of the disk is . So the area is and the volume is
Find the volume of the solid of revolution obtained by rotating the region under the graph of for about the -axis.
Solution
Since is a function of , we chop up the -axis, draw a vertical Riemann sum rectangle and see it rotates into a disk as shown.
With , and , the volume is