3 Ways to Find the Area Under a Curve (and why you would want to do that)

I’ll be honest. I’ve been doing some serious calculus stuff for more years than I would like to admit. Sometimes in physics, you just have to integrate stuff. It’s what happens.

However, when I was helping my son with his calculus class I started to realize — what the heck is going on with integration? I feel like I have a really solid understanding of derivatives — but integration has some weird things.

Well, you aren’t ever going to understand stuff until you break it down to its basic pieces. So, that’s what I’m going to do. I’m going to find the area under a curve three different ways.

Area using summation notation

This is the easiest to understand, but the most difficult to calculate. Oh wait! Why are we finding the area under a curve? OK, hold on — my example at the end should help.

So, the basic idea is to break an area into N rectangles (where N is just some number). I can then find the area of each rectangle and add them up to get the total area. The more pieces I break it into, the better the answer — so I will write an expression for this area and take the limit as the number of pieces goes to infinity.