Upon completion of this unit, students will be able to explain and illustrate the connection between the early attempts to measure non-rectangular and curvilinear objects to the formal methods of elementary calculus. Students should be able to describe social and political influences upon the people using these early methods and how the times in which they lived impacted the development of the fundamental theorem.

Connections: Recognize, use and learn about mathematics in contexts outside of mathematics. This is a study of the historical attempts to solve problems before the fundamental theorem was known. Knowing about the people and their efforts as well as the influences of the era in which they lived can provide an important connection of mathematics to history.

Students should have experienced the use of rectangles or other elementary shapes to estimate area of regions bounded by curves. Knowledge of the historical context for the more famous applications of these methods is not necessary. Students will investigate several persons famous in the history of mathematics. These investigations may lead to others not specifically mentioned in the student activity. it may be helpful to construct a timeline showing where and when some of these people lived and worked.

This lesson requires computers and access to the internet, but no special materials.

Understanding the history of this concept, the fundamental theorem of calculus, should include a knowledge of early calculation methods and the use of rectangles or other basic shapes to approximate area of irregular figures. The historical context in which these early methods were used involves knowing about the people and the circumstances under which they lived and worked. Students can demonstrate an understanding of this by answering questions posed in the activity. Also, the impact of social and political conditions was both helpful and restrictive to these early mathematicians. Students should be able to articulate the effect of such conditions and generalize in other historical contexts.

To investigate and answer these questions, text and Internet resources should be used. 1. Using the extensive biographical information (available from the links above), determine other mathematicians involved with the definition of the integral and the solution to problems such as finding the area of the circle. What roles did they play? For what mathematical concepts or investigations are they primarily known? 2. How did politics and religion affect the work of mathematicians such as Cauchy, Monge, and Carnot? Was the work of any of these or other mathematicians either delayed or enhanced by their political and religious views? 3. Based on your readings of the working conditions of some of these mathematicians, what were some factors that gave rise to or prevented women from becoming mathematicians and professors? 4. What other problems involved the adding of many very small quantities to achieve an approximation? Who were some of the mathematicians and scientists who used these early methods, preceding the formal and more refined calculus we now use?

Internet access is a requirement. Access should be available to computer stations which make use of a browser and word processor simple for the student.