The Volume of a Watermelon

To introduce the idea of finding the volume of a solid of revolution, have students think about finding the volume of a watermelon.

Calculus Grapher

A nice applet from the PhET project at the University of Colorado enabling you to quickly "sketch" a graph of a particular shape and to see graphs of its derivative and integral.

Introduction to Area Under a Curve

In this applet from the GeoGebra Calculus Applets Project, you enter a function and position endpoints to see what the value of the integral is.

The Area Function

This applet from the GeoGebra Calculus Applets Project provides a way to explore the area function. The default integrand can be changed by right-clicking on the curve.

Accumulated Rate of Change and Antiderivatives

This applet from the Flash Mathlets collection by Barbara Kaskosz includes graphs of a bunch of sample rate of change functions. For each, as you drag a slider, you see what the associated population (i.e., antiderivative) would look like. You can also input your own rate of change function and initial condition. What's more, you can draw your own  function, specify an initial condition, draw what you think the associated antiderivative would look like, and then check your antiderivative by moving the slider to see the applet draw the antiderivative.

Definite Integral as an Area Accumulator

This clear, elegant applet from the Flash Mathlets collection by Barbara Kaskosz shows the integral as an area accumulator for a few simple functions.