Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

This is why it was so important to be able to find the area under an arbitrary curve. Because of its intimate connection to the backward problem, the area problem is not just about area. It’s not just about shape or the relationship between distance and speed or anything that narrow. It’s completely general. From a modern perspective, the area problem is about predicting the relationship between anything that changes at a changing rate and how much that thing builds up over time. It’s about the fluctuating inflow to a bank account and the accumulated balance of money in it. It’s about the growth rate of the world’s population and the net number of people on Earth. It’s about the changing concentration of a chemotherapy drug in a patient’s blood and the accumulated exposure to that drug over time. That total exposure affects how potent the chemo will be, as well as how toxic. Area matters because the future matters.