Area, Volume, and Arc Length
We're going to be doing a lot of problems making fish sticks or pizza slices or linguine pasta out of different foods. Since mathematicians are lazy, and because we should be focused more on our knife skills–we like our fingers–we are going to make some assumptions to save time and space later.
From now on, assume that
- all vertical slices have width Δ x and all horizontal slices have thickness Δ y. That way we don't have to keep saying "take a vertical slice of a guava with width Δ x."
- when we say "the height of the slice is ..." we mean "the approximate height of the slice is ...". Similarly when we say "the area of the slice is ..." we mean "the approximate area of the slice is ..." That way we don't have to keep saying the word "approximately." As much as we like that word, we are going to sound like a Star Wars drone if we keep saying it.
- once you know the area of a slice, we can write down the integral that gives the area of the whole region. We don't have to write "we add up all the approximate regions and take the limit as the number of slices goes to ∞ to get an integral" every single time.
- if a problem asks you to "write an integral expression" for the area of a region, that means you don't need to evaluate the integral. If you want to practice evaluating integrals, by all means, feel free to do them. We're focusing on how to set the integrals in this chapter, which is the most difficult part for most people. We can save ourselves the headache of solving them for another day.