7.3 - Volumes

When an area is rotated around something.

• Volume is the integral of area, so if we can find the area of a cross-section of any object and how it changes, we can find its volume.
• Think about whether the area is a disk or a washer and use the circle area formula ($A=\pi r^2$) to find areas and integrate them.
  Notes for this are hard to take on a laptop. Video here.

There are two types of “slice” that you can take out of a rotated area in the plane:

• A disk, with area $A=\pi r^2$
• A washer, with area $A=\pi (R^2-r^2)$
  Integrating these areas gives you the volume of the total rotated area.