Langara College - Department of Mathematics and Statistics

###
Internet Resources for the Calculus Student - Topics in Calculus

##
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,

please do let
us know and we will add them here.