Many shapes in our world are irregular and complicated. The boundary of a lake or stream can be curved and winding. A stock market graph can be highly erratic, with jumps and dips of various sizes. The path of a fly ball is a smooth arc affected by gravity and wind resistance. Clouds are filled with cotton-like bumps and ripples. Measuring our world requires us to confront a variety of irregular shapes, ranging from graceful curves to sharp, jagged edges.

Measuring Irregular Shapes

Irregular shapes can be measured with a technique called discrete approximation, a powerful method that provides the foundation for calculus and a means by which computers perform calculations. Making a discrete approximation involves representing a continuous quantity through a collection of distinct pieces. We live with such approximations every day. For example, a movie reel is a collection of picture frames, shown to...