INTEGRALS AT WORK

Area, Length, Volume, and Mass

The basic applications of integration are in mathematics and physics. If the curved boundary of a plane region can be described in terms of functions, then the region's area and perimeter can be expressed in terms of integrals. In three dimensions there are integrals for the volume and the surface area of solids with curved boundaries. Integration can also be used to determine the mass of a body where the density of the matter inside it can vary from place to place.

Launching Spacecrafts

What are the factors in putting a satellite into orbit or an astronaut on the moon? Close to Earth, the pattern of air flowing around the spacecraft has an important effect on maneuverablity and speed. You need integration to solve the equations that give the best aerodynamic shape for the craft's body. Everything gets easier on the way up: the air thins out, the pull of gravity decreases, the amount of fuel burned lowers the mass left to move, and you are closer to the target orbit. On the other hand, don't let any of the thousands of pieces of rocket debris puncture your hull! You need an enormous number of very fast integrations to calculate and correct the trajectory "on the fly."

Population Growth

Medical scientists, insurance companies, biologists, and politicians are all interested in predicting the size of various populations, both animal and human. The simplest approximation assumes that the growth in size is always proportional to the current size. That can't happen because after a while there wouldn't be enough food, water, or space for the exponentially increasing number of individuals. So a more realistic model is built, perhaps by including a competing population, be it wolves or bacteria, with dynamics of its own, or by building in a factor like a declining birth rate. All such models use systems of equations that need integration at some point in their solutions.