The Integral
For those of you with a good background in superhero (or Seinfeld) lore, you’ll know what I mean by Bizarro world. In Bizarro world, everything is the opposite of this world; good means bad, up means down, and right means left. Because Superman is smart in our world, Bizarro Superman is stupid. Well, integrals are Bizarro derivatives. Deriving takes us from a function to an expression describing its rate of change, but integrating takes us in the opposite direction—from the rate of change back to the original function.
Even though integrating is simply the opposite of deriving, you might think that its usefulness would be limited. You’d be wrong. There are just about as many applications for integrals as there are for derivatives, but they are completely different in nature. Instead of finding rates of change, we’ll calculate area, volume, and distance traveled. We’ll also explore the Fundamental Theorem of Calculus, which explains the exact relationship between integrals and the area beneath a curve. It’s surprising how straightforward that relationship is and how dang useful it can be.