Various methods were considered by thinkers such as Archimedes (287-212 B.C.). One popular method used the idea of upper and lower limits. If you take the shape you wish to measure and draw a regular polygon that would fit just inside the irregular shape, this gives a lower limit to the area of the irregular shape. Then draw a regular polygon that only just completely surrounds the irregular shape. This gives an upper limit to the shape, so the actual area must lie between the two areas drawn. If one regular shape was not close enough, then several packed together like a jigsaw could be used. While this method could get close approximations, it was in a sense a self-defeating process, as you could never get the correct answer.
Centuries later European mathematicians such as Johannes Kepler (1571-1630) introduced the use of infinitesimals to their calculations of areas. Infinitesimals were thought of as quantities smaller than any actual finite quantity, but not quite zero. While this is a somewhat strange idea, it was very influential. The first textbook on integration methods was published in 1635 by Bonaventura Cavalieri (1598-1647), but in a form quite unrecognizable today. Further modifications were made by various thinkers, each using their own special method for approximating the area under a curve.
This is a free page. This page contains 196 words. This
article contains 1,683 words (approx. 6 pages at 300
words per page).
Read the rest of this Article with our Lebesgue's Development of the Theories of Measure and Integration Access Pass.
