At the Manipula Math with Java site in Japan there are several applets demonstrating how the volume of a Solid of Revolution corresponds to a limit of Riemann sums. The JAVA applet allows a curve to be rotated about the x-axis. Cross-sections can be highlighted and shifted, and the number of disks making up the volume increased or decreased, and then the cross-sectional area and the volume are calculated.
Our 'raw list' (of links provided without comment) includes many more examples of sites demonstrating applications of integration, including more volumes, arc length, some applied sciences, and probability and statistics.
In many of these examples, once we have used the Riemann sum idea to see that what we want is really an integral, that integral is then most easily calculated by using the Fundamental Theorem of Calculus to reduce it to an antiderivative problem.
But sometimes we really do want to go the other way. For example, 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? This idea is explored in the page on Accumulating
Rates of Change, in the Gallery
of Interactive Geometry, at the University of Minnesota's Geometry
If you have come across any good web-based illustrations of these and
please do let us know and we will add them here.
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