The Visual Calculus App
What it teaches: This app describes visually the processes of differentiation, indefinite integration and definite integration (the area under the curve).
How it works:
Visual Differentiation: You can draw a freehand curve or choose from a number of predefined curves. This curve is then digitized (broken into 10 or 15 straight line segments). When you click on differentiate the segments fall one-by-one onto the x axis and produce the derivative. Here is an example of the display after the animation is finished.
Indefinite Integration: You can draw a freehand curve or choose from a number of predefined curves. This curve is then digitized (broken into a number of rectangles). When you click on integrate the segments rise one-by-one and arrange themselves end-to-end (i.e. integrate themselves) to produce the indefinite integral. Here is an example of the display after the animation is finished. Notice the movable pointer which represents where the indefinite integral starts.
Definite Integration or Area Under the Curve: This is exactly the same as the indefinite integration animation except that it continues with another animation in which the pieces of the integrated function move to the right to form a stack whose height represents the area under the original curve. Here is an example of the display part way through the animation:
... and after the animation is finished:
Some ideas for teachers:
- In a regular calculus class students learn quite quickly that, for example, the integral of x2 is ⅓x3+C, but they have problems understanding how to obtain the integral of a piecewise-defined function. What this app makes clear is that you don't have to integrate a formula; you can integrate any curve that you can sketch freehand. The method is always the same – you join the pieces end-to-end.
Where to now?