Calculus is a branch of mathematics that has found numerous applications in the sciences since its introduction at the end of the seventeenth century. It is the language which engineers, physicists, and other scientists use to express their theories in mathematical terms and to solve practical problems. For example, most of the laws of physics are expressed in terms of calculus.
Calculus was invented to answer questions that could not be solved using algebra or geometry. The branch of calculus that we study in MATH 140, called differential calculus, began with questions about the speed of moving objects. How fast does a stone fall two seconds after it has been dropped from a cliff? Originally, differential calculus was developed to solve problems connected with the motions of the planets. The English scientist Sir Isaac Newton and the German philosopher Gottfried W. Leibniz independently laid down the principles of calculus. More generally, differential calculus is concerned with rates of change. Speed can be viewed as the rate of change of distance along a line with respect to time.
Another branch of calculus, integral calculus, which is studied in MATH 141, was invented to answer a very different kind of question. That of finding the area of shapes with curved sides? Initially it deals with the question of finding a function when its rate of change is known. This leads to the concept of the integral which is analogous to the sum of a set of numbers. But while a sum is taken over a finite discrete set, the integral is taken over a continuum, for example a bounded interval. Whenever an engineer wants to calculate a continuous sum, for example the lift on an aircraft wing, where the pressure ( force per unit area) varies from place to place, he will need to evaluate an integral.