These directions should assist in the construction of a trapezoidal estimate of the area under a function f(x) on an arbitrary x-interval (a,b).

1:
Begin by plotting the new function f(x) = (x^2)/8. A function that is linear or very steep will not be helpful in the trouble-shooting of this type of construction. The function can be changed after the construction. Create a parameter (t[1] in the diagram) and give it a value of 5. This parameter will be used to define the number of sub-intervals of area. The value can be changed later to produce a better-fitting area estimate, but if it is too big, the widths will be inconveniently small during the rest of the construction process. Create a second parameter equal to t[1]-1. This will be the number of times that we want to iterate the first trapezoid, since the original trapezoid will not be part of the iteration. Create a third parameter equal to 1/t[1]. This will be the width of each interval. Next, create two independent points (A and B in the diagram below).

2:
Select both points A and B and the x-axis and construct parallel (horizontal) lines through the points. Next, select both points and the x-axis and construct perpendicular lines through the points, as shown. Create the intersection C (as shown) and dilate that point by a factor of 1/t[1] with A as the center. Mark the vector from A to the dilated point. Now create a point D on the vertical line passing through A and translate D by the marked vector. D is the point we will iterate, and any structure dependent on D will be iterated to the specified depth.

3:
Select D and the translation of D (Sketchpad will call this D') and go to "Measure" - "Abscissa (x)". It will create two new measurements, the x-values of each point. Now "Calculate" f(x[D]) and f(x[D']). Plot the two points ( x[D], f(x[D]) ) and ( x[D'], f(x[D']) ). Create a new parameter with value 0 and plot the two points ( x[D], 0 ) and ( x[D'], 0 ). These four plotted points are the vertices of our first trapezoid.

4:
Create a quadrilateral interior using the four plotted points. Now select the point D and the parameter t[1]-1. Click "Transform", hold down the shift key and select the "Iterate To Depth" option that will appear. The pre-image D should be sent to D' in order to construct the iteration. I generally go into the structure menu and tell it to iterate "Non-Point Images Only" and I uncheck "Tabulate Iterated Values".

5:
Now the construction is in place and the function can be changed. The number of intervals can be changed (just change t[1] and the other parameters will respond accordingly) to give a smoother picture.

6:
To shade the area between two functions, use the points ( x[D], g(x[D]) ) and ( x[D'], g(x[D']) ) instead of ( x[D], 0 ) and ( x[D'], 0 ).

7:
One of the most common mistakes that I make is to iterate A, the point which determines the width of the interval. As the width of each trapezoid is a fraction of the horizontal distance between A and B, this will decrease with each iteration. First it will use |AB|/t[1], then |A'B|/t[1], then |A''B|/t[1]... and the iterated image will never get all the way across the interval. Instead iterate D, which will not change the distance between A and B, so the intervals will have constant width.

Note:
This method creates an area that stretches and conforms predictably to any adjustment of the axes. I have made tools for this (available in some of the sketches on this page) and it should be noted that there are some complications regarding the square/rectangular axes when you try to generalize this process as a tool. This is because one sketch may contain multiple coordinate systems.