You’ve learned how to calculate integrals. Now comes the fun part — using them to answer real questions. How long is a curved wire? What force does water exert on a dam? Where is the balance point of a metal plate? How much value do consumers gain from a market? What’s the chance that a measurement falls within a certain range? Integration lets us add up infinitely many tiny contributions, and that’s exactly the key to all these problems. In this chapter, we’ll set up integrals to measure lengths, areas, forces, economic surplus, and probabilities.