If we take an even number of points n between a and b, we can approximate every 2 intervals by a parabola, by finding the (unique) parabola going through the f values of the 3 points in each closed interval. We can then approximate the integral by integrating each parabola on each interval. The expression we get when we do this is[br][br]Area = Δx/3[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + .... + 2 f(x_{n-2}) + 4 f(x_{n-1}) + f(x_n)].[br][br]As n grows, this converges to the integral much faster than the midpoint or trapezoid approximations.

How to use this demo:[br]- Drag a and b along the x-axis to change the left- and right-hand sides of the definite integral.[br]- Change n by dragging the slider back and forth to change the number of boxes in the approximation.[br]- Type "f(x) = x^2 + 1" or any function of your choice into the Input Bar to run the demo for other functions.[br]- Click the checkboxes to view different approximations of the integral.