This applet shows three points on a curve separated in [math]x[/math] by [math]h[/math]. Straight dotted lines connect the points.[br]Moving the blue dot will change the [math]x[/math] location.[br]Moving the black dots will change the function.[br][br]There are three check boxes.[br][list][br][*]Trapezoids shows the areas that would be used for trapezoid numerical integration.[br][*]Parabola shows a quadratic equation fit through the three points.[br][*]Simpson's Rule shows the area that would be used for Simpson's rule integration.[br][/list]

Trapezoids[br] Note the area between the trapezoids and the curve. This would result in an error in the approximation.[br] How does this area vary as [math]h[/math] is decreased?[br]Parabola[br] How close is the parabola to the functions curve?[br] How does this compare to the straight lines?[br] Is the comparison the same as you move [math]x[/math] and change [math]h[/math]?[br]Simpson's Rule[br] How does the area for Simpson's rule integration compare to the actual area under the curve?[br] Does it appear that Simpson's rule area or trapezoid method would better approximate the true area under the curve?