Langara College - Department of Mathematics and Statistics

## Fundamental Theorem of Calculus

The connection between the integral (or area) and derivative (or slope) problems is known as the Fundamental Theorem of Calculus.  It is important not just as a tool for quickly finding some
integrals, but also for helping us to recognize the relationship between processes of change and
accumulation in many applied fields. This theorem is what enabled the rapid advances in our
understanding of the physical world that underlie the industrial revolution. Its discovery is without a doubt one of the most significant events in all of human history!

This Online Lab from UBC  covers both the rectangular approximation to area and the Fundamental Theorem. (You won't be able to "hand in" your results but can do the activities and look at the posted solutions).

This Cumulative Area Applet  (byDanSloughter@FurmanUniversity) allows you to see how the "area so far" changes as you move along the graph of a particular function. (Note that when the author refers to the "area" he really means the net area - ie area above x-axis minus area below).  And if you want to draw your own graph then you can use this applet from Austria and see by eye that for whatever graph you draw (in red), their blue graph looks right for both an antiderivative and for the net area-so-far)

This applet from the IES group in Japan allows you to look explicitly at the ratios of change in area to change in position and to explore how they approach the value of f(x) as the change in x goes to zero.

See also Numerical Integration : Accumulating Rates of Change, at the Gallery of Interactive Geometry, from the University of Minnesota's Geometry Center. At this site it is possible to explore the numerical integration of data sets. If the velocity of an object has been measured at certain instants of time, is it possible to "integrate" this discrete data to estimate the change in the object's position?

You might also check our 'raw list' (of links provided without comment) to see if there are any more examples there that we haven't yet included here.

If you have come across any good web-based illustrations of these and related concepts,