## IntTheorems Applet

IntTheorems is used in demonstrating a wide range of definite integral theorems. It is configurable through a choice box as to which theorem is being illustrated. The options are:

- Zero Rule and Reverse Limits: illustrates what happens when both limits of integration are the same, or when the upper limit is smaller than the lower limit.
- Constant Multiplier: illustrates what happens when the integrand is multiplied by a constant.
- Addition and Subtraction: illustrates the addition and substraction of two definite integrals with the same limits.
- Internal Addition: illustrates the addition of two definite integrals with the same integrand and which share one of the limits.
- Domination: shows that if one integrand is always greater than another, then the value of the integral of the first is always greater than the second (assuming common limits).
- Min/Max inequality: shows that the value of a definite integral is between the minimum value of the integrand over the interval, times the width of the interval, and the maximum value of the integrand over the interval, times the width of the interval.
- Area between curves: illustrates the concept of the area between two curves.

The applet tag that creates the applet is as follows:

<applet code="net.sourceforge.webcompmath.calculus.IntTheorems" codebase="../../assets"

archive="webcompmath.jar,calculus.jar" height="480" width="640">

<param name="UseLoadButton" value="no" />

<param name="Example1" value="1. Zero Rule & Reverse Limits;0;-1,5,-2,40;e^x;0;0;1;-1;5;3;-1;5" />

<param name="Example2" value="2. Constant Multiplier;1;-1,5,-10,10;x;0;c*f(x);1;-1;5;3;-1;5;2;-5;5" />

<param name="Example3" value="3. Addition;2;-1,5,-2,30;x;e^x;f(x)+g(x);1;-1;5;3;-1;5" />

<param name="Example4" value="4. Internal Addition;3;-0.5,5,-2,30;e^x;0;0;1;-0.5;5;3;-0.5;5;2;-0.5;5" />

<param name="Example5" value="5. Domination;4;-0.5,5,-2,30;1/2*e^x;e^x;0;1;-0.5;5;3;-0.5;5" />

<param name="Example6" value="6. Min-Max Inequality;5;-0.5,5,-2,30;e^x;0;0;1;-0.5;5;3;-0.5;5" />

<param name="Example7" value="7. Area Between Curves;6;-1,5,-5,5;x;0.5x;0;1;-1;5;4;-1;5" />

<param name="Example8" value="8. Area Between Curves, More Complicated;6;-1,5,-1,3;0.5x;sin(x);0;1;-1;5;4;-1;5" />

</applet>

This applet takes the following parameters:

Param NameTypeDefaultDescriptionExample1, ...String N/A Each example is a list of strings separated by semicolons. The strings, in order, are:

1. The name of the example, displayed in the example menu.

2. The setting for the choice box, which controls what gets shown on the graph and what kinds of controls are displayed. The values to use are: "0" for zero rule and reverse limits; "1" for constant multiple; "2" for addition and subtraction; "3" for internal addition and substraction; "4" for dominance; "5" for min/max inequality; "6" for area between curves

3. The expression for f(x).

4. The expression for g(x). Note that if this is not used in a particular configuration for the applet, you must still define a function (e.g., "0") otherwise you will get a "Can't parse empty string" error)

5. The expression for h(x). Like g(x), must be defined as something, even just "0", if not being used.

6. The value for a.

7. The minimum value for the a slider.

8. The maximum value for the a slider.

9. The value for b.

10. The minimum value for the b slider.

11. The maximum value for the b slider.

12. The value for c.

13. The minimum value for the c slider.

14. The maximum value for the c slider.