GROUP PROJECT  FUNDAMENTAL THEOREM OF CALCULUS --- INCLUDES BOTH THEOREM. BY TSHERING YANGZOM AND LANNY NG

Newton discovered his fundamental ideas in 1664– 1666, while a student at Cambridge University. During a good part of these years the University was closed due to the plague, and Newton worked at his family home in Woolsthorpe, Lincolnshire. However, his ideas were not published until Leibniz, in France and Germany, on the other hand, began his own breakthroughs in 1675, publishing in The importance of publication is illustrated by the fact that scientific communication was still sufficiently uncoordinated that it was possible for the work of Newton and Leibniz to proceed independently for many years without reciprocal knowledge and input. Disputes about the priority of their discoveries raged for centuries, fed by nationalistic tendencies in England and Germany. Discovery of the Fundamental Theorem of Calculus

The first part of this theorem tells us how to evaluate a definite integral provided that f has an indefinite integral. The second part of the theorem tells us that f has an indefinite integral. The only remaining problem is actually finding a formula for the indefinite integral which we can easily evaluate.

Example 1 Notice we took the constant of integration, C, to be 0.

Applying the Fundamental Theorem of Calculus:

Let’s look at example 1 again, but using some useful new notation which streamlines this process:

This is the area of one hump of the sine wave: Example 2

The result of this calculation is somewhat surprising!

CITATION ath6A/Lectures/ch5/3/FundamentalT heorem.htm calculus/4/ftc.9/