Area, Volume, and Arc Length
We know how to integrate our favorite flat foods to find their areas. Our next step is to learn how to integrate our favorite 3-D foods to find their volumes. In terms of the methods we will learn, there are two types: solids with known cross-section and solids of revolution. With these methods, we can find the volume of any 3-D food we want. 4-D foods like the hypergeometric cotton candy will have to wait.
A solid with known cross-section is like a loaf of cinnamon raisin bread. There may be a lot of things going on within each delicious slice. We are going to slice it that way, nonetheless. The base of the solid is some 2-D region. For this loaf of bread, we can think of the base as a circle in the xy-plane:
When we slice the loaf perpendicular to the x-axis, we get slices that look like semi-circles. The size of the semi-circle depends where along the base the slice was cut. Slices in the middle of the loaf will be bigger and have more cinnamon than slices at the ends. We only care about the area of the slice.
A solid of revolution is a little different from cinnamon raisin bread. Let's imagine a very, very thin piece of bundt cake. That slice, flavored like key lime in this case, is just a 2-D area with a lot of flavor. We could settle for this thin slice, but we want more. With the cake in front of us, we rotate our knife around until we sweep out the volume we want, and then we cut out a large piece. That piece has the area of our thin slice but a much larger scrumptious volume for us to enjoy.
A couple other good examples of solids of revolution include party paper bells.
To find the volume of a 3-D solid, we will use a similar attack method to the one we used to find the area of a 2-D region.
1) Understand the solid whose volume we're finding.
2) Slice the solid and find the approximate volume of each slice. The difference between the two methods, solids with a known cross-section and solids of revolution, involves slicing.
3) Add up the volumes and take the limit as the number of slices approaches ∞ to get an integral.
All the assumptions we made for finding area still apply if we replace the word "area" with the word "volume." Well, except the first one–when we get into three dimensions we don't really have "vertical" and "horizontal" slices anymore.