Partition, Differential, Boxes
 5  PARTITION & SUMS 4 boxes
 Links
 Hide/Show Boxes
 Hide/Show g(x)
 Hide/Show lefthand sums
 Hide/Show g(x)
 Go to
ReimannSums.gsp

 Notes
 *
Reimann.htm exists and is formated like this page and includes
ReimannSumNotes.pdf and
ReimannSums.gsp
 * The next page shows example of area and sum computation.

 Activities
 * Show boxes.
 * Discuss and position lefthand boxes, righthand boxes, midpoint boxes.
 * Discuss the areas and sum of area of
lefthand boxes, righthand boxes, midpoint boxes.
 * Move boxes from picture.
 * Change a and b and perhaps f(x).
 * Show boxes and discuss areas.
 * Move boxes from picture.



 6  REIMANN & SUMS
 Links
 *
ReimannSumNotes.pdf
 *
Reimann.htm including
ReimannSums.gsp

 Activities
 * Discuss areas, summation meaning & format
 * Perhaps go to links



 7  SUMS f(x)  g(x), slide end points to change [a, b]
 Links
 Show height and area computation
 Show midpoint coputation
 Show g(x)

 Activities
 1. Start w/g(x) = 0
 2. Ask can an area be negative?
 * Show negative & positive areas.
 * Show the left vs midpoint vs right turning negative as x is dragged.
 3. Ask how would increasing the number of boxes effect the area?
 4. Ask which is the best approach for computation & why?
 5. Reveal g(x) and discuss height.
 6. Change f(x), g(x), or both.
 7. N is small (so the differences in the 3 different sums is large).
Use the midpoint sum to answer this question.
As is, the sum is an approximation of the area under f(x) from a to b. What is the result if one wishes the sum from b to a?



 8 SUMS f(x) g(x), input [a,b] in the boxes for a and b
 Note
 This page only provides the leftsum, but provides a
more accurate way to name a and b.

 Links
 Show g(x)

 Activities
 * Change f(x) to an odd function and ask questions.
 * Change f(x) to an even function and ask questions.



 9  CUMULATIVE AREA probability distribution
 Notes
 My calc students don't usually take statistics because they are taking calc.
 This sheet serves as a background and shows a real life example of using an integral.
 N is 32. This is small, compaired to infinite, but large enough to provide a good estimate.

 Two images are provided. The top is what the sheet usually looks like.
The bottom shows notes.

 This also illustrates the use of the term "cumulative" to indicate the integral
of a function and provide/state cumulative results.

 Links
 * Show function f(x, mu, sigma)
 * Show constant 1.
 * Show constant 2.
 * Download statistics spreadsheet  use sheet cum
This has an integration function feature.
 * Link to "Probability for Calc I"
This provides a summary of statistics and the normal probability distribution.


 Activities
 * Play.



 10  HISTORY sum 2 integral
 Links
 1st. Symbols & vocabulary
 2nd: Exactly how do they relate?
 3rd: How Does the finite go to the infinite?
 4th: FTC I and II
 Activities
 * In order reveal and discuss the notes.



 11  INTEGRATION by dots
 Links
 FTC I and II
 Plots on the next page

 Activities
 * In order reveal and discuss the notes.



 12  INTEGRATION FTC I & FTC II
 Notes: The plotted points are off by a constant. It is an indefinate integral.
Change the parameter "plus c" to adjust the plotted antiderivative points.

 Links
 * Show g(x) and height = h(x)
 * Show ordered pairs of plotted points.
 * Show "Why plus c?"
 * Show blue Suggested Functions box.

 Activities
 * Change the function as desired.
 * Functions that fit on the screen are suggested.

