Dynamic Calculus

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A rectangle of differences

The applet below shows the area bounded by the curve \(y = f(x)\), the \(x\)-axis and the ordinates \(x = 2\) and \(x = 8\). Investigate the limit of the sum of the upper rectangles and the limit of the sum of the lower rectangles and determine their relationship with the area under the curve.
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]

 

1. Move the first slider to the left and then all the way to the right. What generalisation can you make regarding the accuracy of our area approximation as the number of sub-intervals, \(n\) increases?
2. Move the second slider all the way to the right. A 'rectangle of differences' is constructed which equates to the difference between the areas of the upper and lower rectangles. What do you notice about the dimensions of this rectangle?
3. As \(\delta x\) approaches zero (ie the number of sub-intervals increases), what can be said about the base of the rectangle? Does the height of the rectangle change? What does the area of the rectangle approach as \(\delta x\) approaches zero?
4. What does your answer to (3) indicate about the limit of the sum of the lower rectangles and the limit of the sum of the upper rectangles?

State of New South Wales, Department of Education and Training 2008, Created with GeoGebra