Dynamic Calculus

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Upper and lower rectangles

How do we calculate the area bounded by the curve \(y = f(x)\), the \(x\)-axis and the ordinates \(x = a\) and \(x = b\)? This applet approximates this area by taking the sum of the areas of the upper rectangles and by taking the sum of the areas of the lower rectangles.
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]

With the 'slider' at position \(n = 1\), click the checkboxes for the sum of the lower rectangles and the sum of the upper rectangles. Describe the relationship between: the area \(A\) under the curve; the area of the lower rectangle; and the area of the upper rectangle.
Increase the number of sub-intervals to 2. Is the relationship you described in question (1) true for \(n = 2\)? Is the relationship true for other values of \(n\)?
Click the 'difference' checkbox. What do you observe as \(n\) increases?
The exact area has been calculated and can be viewed by clicking the 'exact area' checkbox. As \(n\) increases, what observation can you make regarding the approximations of the area under the curve using the upper and lower rectangles?
Keeping \(n = 20\), predict what will happen to the value of the 'difference' as the point \(a\) is moved closer to the point \(b\). Justify your prediction.