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This applet gives a simple illustration of the Riemann sum idea as it applies to calculating the area under a curve.

Here, the area under a parabolic curve is approxomated by adding the areas of rectangles.

In the first picture, the rectangles all sit below the curve, and the sum of their areas is an underestimate of the area under the curve.

In the second picture, the rectangles more than cover the area under the curve, and the sum of their areas is an overestimate of the area under the curve.

By moving the slider, the number of rectangles is changed, as are the resulting estimates.