6.2 - Volumes

Another fairly short section, since this is the last section in 3D.

The chapter starts off reminding us what a cylinder is, in case you slept through middle school. They also introduce the term "Rectangular parallelepiped" which is a really pretentious way to say "box" because too many people started the chapter knowing what they were talking about.

But if we don't have an easy shape like a cylinder where we can just figure out the information we need from the axes, and instead have the weird-looking slug thing they want us to find the area, we do it similarly to how we find areas, only with cross-sections instead of rectangles. When we slice out a cross-section, we can call it a cylinder with the area of the section and a height of ▲x. Whatever ▲x we choose, if we add up all the cross sections, we get an approximation of the volume the object.
The approximation gets better and better as the number of cross-sections get higher, so, it stands to reason that, just like with our rectangles, the limit of the approximation as the number of cross sections approaches infinity is the volume.

The rest of the chapter just shows a few example problems, as well as proving with Calculus that the volume of a sphere is (4/3)πr^3. The only other main idea the examples show is that if, when rotating around the given axis, you get a shape with a hole in it, you just subtract the volume of the empty spot (found by treating as its own object) from the volume of the outer object.
Then, if the shape cannot be cut into disks, cut it into shapes that work. They showed a square-bottomed pyramid and a wedge cut out of a cylinder. Basically, find how to figure out the volume of a cross section, and use it instead.

Fairly short. Not too bad. The equations they used for the non-cyllindrical shapes were a bit confusing, but I think I'll figure out while working out some of the problems.